拓扑空间、开集、闭集、闭包、聚点、邻域

拓扑空间、开集、闭集、闭包、聚点、邻域 第一章 拓扑空间与拓扑不变量 数学分析中的连续函数的定义与和值域都是欧氏空间(直线、平面或空间)或是其中的一部分。本章将首先把连续函数的定义域和值域的主要特征抽象出来用以定义度量空间，将连续函数的主要特征抽象出来用以定义度量空间的连续映射。然后将两者再度抽象，给出拓扑空间和拓扑空间之间的连续映射。随后逐步提出拓扑空间的一些基本问题如邻域、开集、闭集、闭包、聚点、导集、内部、边界、序列、极限等。进一步引入紧致性、连通性、可数性与分离性等重要的拓扑不变性 ?1.1拓扑空间、开集、闭集、聚点、闭包、邻域 一、问题的引入 数学分析里我们知道，在连续函数的定义中只涉及距离这个概念，定义域是一维欧氏空间,即实数空间，两点之间的距离d(x,y)=|x-y|,即两两实数之差的绝对值,定义域是n维欧氏空间，两点x=(x,x,„，x),Y=(y,y,„，y) 之间的距离 1 2n12n 22()+(xy)xy,，,…d(x,y)= 。 11nn 无论是几维空间，它的距离都有下面的性质: nR, 1. d(x,y)?0 , x,y? ; 2. d(x,y) = 0 x = y ; , nnRR,, 3. d(x,y) = d(y,x) x,y? ; 4. d(x,z) ? d(x,y) + d(y,z) , x,y,z? ; 这些性质反映了距离的特征。 nR将推广为一般的集合，我们由距离可以抽象出度量以及度量空间的定义。 (一 ) 度量空间 1. 定义 定义1 设X是一个集合，ρ:X×X?R ,如果对于任何x,y,z?X,有 ?(正定性)ρ(x,y)?0 并且ρ (x,y) = 0 x = y ; , ?(对称性)ρ (x,y) = ρ (y,x) ; ?(三角不等式)ρ (x,z) ?ρ (x,y) + ρ (y,z) 则称ρ是集合X中的一个度量。 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 如果 ρ是集合X中的一个度量，则称偶对(X，ρ)是一个度量空间，或 径称X是一个度量空间。而ρ(x,y)称为从点X到点Y的距离。 2. 度量空间举例 例2.1.1 实数空间R 对实数集合，定义ρ:R×R?R如下:x,y?R，令ρ(x,y)=|x-y| ,易知ρ是R, 的一个度量。因此(R,ρ)是一个度量空间。 可见，度量空间是实数空间的推广，度量是距离的推广。 nR例2.1.1 n维欧式空间 nnnRRR对实数集合R的n重笛卡尔积=R×Rׄ×R，定义ρ:×?R如下: nn2R对任意两点x=(x,x,„，x),Y=(y,y,„，y) ?，令ρ(x,y)= ， (xy),1 2n12n,iii1, nnnRRR可以验证ρ是的一个度量，偶对(，ρ)称为n维欧氏空间。有时径称为 2n维欧氏空间。n=2时，R常称为欧氏平面或平面。 例2.1.2 Hilbert空间H 记H是平方收敛的所有实数序列构成的集合，即H={ x=(x,x,„，x) | x ?1 2ni ,2R, i?Z , },定义ρ:H×H?R如下:对于任意x=(x,x,„，x,,+1 2,ii1, ,2x),Y=(y,y,„，y) ?H，令ρ(x,y)= 。这个定义的合理性及验(xy),n12n,ii,i1 ,2,,证以及验证ρ是H的一个度量，可见P 附录。因此(H, ρ) (xy),49,ii,i1 是一个度量空间，称为Hilbert空间。 例2.1.3 离散的度量空间 设(X, ρ)是一个度量空间，称(X, ρ)是一个离散的度量空间或称ρ是一 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 ,个离散的度量，如果对每一个x?X,存在一个实数使得ρ(x,y)> ,对,,0xx 任何y?X，y ? x成立。 如，设X是一个集合，定义ρ:X×X?R ,使得对于任何x,y?X,有 0若xy,, , 易知ρ 是X的一个离散度量，度量空间(X, ρ)是,(,)xy,,1若xy,, 离散的。 思考题 例2.1.5 令X= C ([a,b]) = {f: [a,b]?R |f在[a,b]上连续}，并且对于任意的f , g b?C ([a,b]),令d(f,g)= , d是C ([a,b])的度量吗, |f(x)-g(x)|dx,a (答案: d是C ([a,b])的度量，因此(C ([a,b])，d)是一个度量空间) 3. 邻域、开集 ? 度量空间的球形邻域及其基本性质 定义2. 设(X, ρ)是一个度量空间，x?X, 对于任意的ε>0， B(x, ε)={y?X |ρ(x,y)< ε} 称为以x为中心，ε为半径的球形邻域， 也称为x的一个ε邻域，也记作B(x) 。 ε 定理1.0.1 度量空间(X, ρ)的球形邻域具有以下性质: ? 每一点x?X至少有一邻域，并且x属于它的每一个邻域; ? 对于点x?X的任意两个球形邻域，存在x的一个球形邻域同时包含于 两者; ? 如果y?X属于x的某个球形邻域，则y有一个球形邻域包含于x的那个球形邻域。 证明: „ „ ? 度量空间的开集及其基本性质 ,,，,aA,0都,定义3. 设X是一个度量空间，AX，如果，使B(a, ε) , X ，则称A是X的一个开集。 , 由定理2.1.1的?知～X的球形邻域都是开集。 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, a3 part from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 例2.1.7 实数空间R中的开区间都是开集，而半开半闭区间、闭区间都不是开集。两个开区间的并也是开集。 可见，度量空间的开集是实数空间开区间的推广。 定理1.0.2 度量空间X的开集具有以下性质: ? 集合X本身和空集Ф都是开集; ? 任何两个开集的交是开集; ? 任何一个开集族的并是开集。 证 „ „ 推论 U是度量空间的开集的充分必要条件是U是这个空间中若干个球形邻域的并。 ? 度量空间中点x的邻域---球形邻域的推广 定义4. 设X是一个度量空间， x?X,UX，如果存在开集V使x?V ,,U ，则称U是x的一个邻域。 注: 有定义可知～开集V是它的每一点的邻域～但邻域却不一定是开集。如[0～2]是1 的邻域～但它不是开集。 定理1.0.3 设X是一个度量空间，x?X,UX，则U是x的一个邻域存,, 在B(x,ε) U。 , 证明:„ „ 本定理为邻域提供了一个等价说法。 推论 X是一个度量空间～ UX～ 则U是X的一个开集U是其内每一点,, 的邻域。 由定义2.1.3和定理2.1.3 。 证 (二 ) 度量空间之间的连续映射 定义5 设X和Y是两个度量空间，f:X?Y，以及x?X，如果对于f (x)00的任何一个球形邻域B(f(x),ε)，存在x的某一个球形邻域B(x，δ)使得f (B(x，0000δ)) B(f(x),ε)，则称映射f在x 处是连续的。 ,00 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 如果映射f 在X的每一点连续，则称f 是一个连续函数。 显然这个定义是数学分析中连续函数定义纯粹形式上的推广。 定理1.0.4 设X和Y是两个度量空间，f:X?Y，则 ? f在x 点处连续 f (x)的每一个邻域的原像是x的一个邻域; ,000 ? f 是连续的 Y中每个开集的原像是 X中的开集。 , 证明: ? “”若f在x 点处连续，设U为f (x) 的一个邻域，据TH2.1.3， ,00 有B(f(x),ε) U，因为f在x 点处连续，所以存在B(x，δ)使得f (B(x，δ)) ,0000 -1 -1-1 B(f(x),ε)，然而f[B(f(x),ε)] f(U)，而B(x，δ) f[B(f(x),ε)]，,,,0000 -1-1所以B(x，δ) f(U)，这说明f(U)是 x的一个邻域 。 ,00 “” 设f (x)的每一个邻域的原像是x的一个邻域，任给f (x) 的一个邻,000 -1 域B(f(x),ε)，则f[B(f(x),ε)]是x的一个邻域，据TH2.1.3，x有一个球形0000 -1 邻域B(x，δ) f[B(f(x),ε)]，因此f[ B(x，δ)] B(f(x),ε)，所以f,,0000在x 点处连续。 0 -1 “?”设f连续，令V为Y中一开集，U= f(V),对于每一个x?U，, 则f(x) ?V，由于V是开集，所以V是f(x)的一个邻域，由于f 在每一点x连续，故由?知U是x的一个邻域，由上面的推论知，U是开集。 “”设Y中每个开集的原像是 X中的开集，下证f 在任一点x?X连续。, -1,,设U是f(x)的一个邻域，即存在开集V使f(x) ?VU，从而x ?f(V) -1-1-1f(U),由条件f(V) 是 X中的开集，所以f(U) 是x的一个邻域，于是?中必要条件成立。所以f 在点x?X连续。由于x的任意性，所以f 是连续映射。 5 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 二、拓扑空间、开集、闭集 参照度量空间中开集的基本性质(TH1.1.2) 建立拓扑空间 定义1.1.1 设X是一个集合，T 是X的一个子集族，如果T 满足如下条件: ? X ,Ф?T ; ? 若A,B?T ，则A?B?T ; A,T? 若T T ，则 。 ,1 : A,T 1 则称T是X的一个拓扑。 若T是X的一个拓扑～则称偶对(X, T )是一个拓扑空间，或称集合X是相对于拓扑T 而言的拓扑空间;或T 不需指出时，径称集合X是一个拓扑空间。T 中每一个元素叫做拓扑空间(X, T )或X中的一个开集;开集的补集称为闭集。 说明:? 条件?蕴含着:当n > 1时若A ，A ，„ „， A?T ,则A12n 1?A?„ „?A?T 。(但对无限交不一定成立，见后面的例) 2n ? ?、?两条常被称为关于有限交、无限并封闭; A,,,T? 当T =Ф时，, 这一点在?中已有，因此以后验1 :A,T 1 证?成立只需对T?Φ验证即可,1 ? 有拓扑空间的定义和度量空间开集的基本性质知，度量空间都是拓扑空间。关于这一点还有下面的定义: 定义1.1.2 设(X, ρ)是度量空间。令T 是由X中的所有开集构成的集ρ 族，据TH1.0.2，T 是X的一个拓扑。我们称T为X的由度量ρ诱导出来的拓ρρ 扑。 约定:说度量空间(X, ρ)的拓扑时，如果没有另外说明，就指T ，称ρ其为拓扑空间时就指(X, T) 。 ρ coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 n2因此，实数空间R，n维欧氏空间R (特别，欧氏平面R),Hilbert空间H都可以叫做拓扑空间，其拓扑就是其各自的通常度量诱导出来的拓扑。 1111在实数空间中，()是开集，但不是开集。:aaa,，,aa,，,(,){}nZ,，nnnn 这说明无限个开集的交不一定是开集。 定理1.1.1 设X是一个拓扑空间，记F 为所有闭集构成的集族。则: ? X,Ф?F ; ? 如果A,B?F ，则A,B?F ; A,F? 如果Ф?F F ，则 。 ,1:A,F1 证明 ? 由于X,Ф?T ，所以Ф=X′，X=Ф′?F 。 ? 当A,B?F 时，有 A′,B′?T ，从而A′?B′?T ，因此 A?B = A〞?B〞=(A′?B′)′?F 。 U,T ? 令T ={A |A′?F },于是T , T ，因此，从而111:U,T1 '',,,AAAU()()F,,,, 。证毕。 ::::AAAU,,,,FFFT1111 注:? ?蕴含着～n>1时～A～A～…～A 是闭集～则A?A?…?A 也是12n12n闭集。即闭对有限并封闭, A? ?中要求F ?Ф～因为F =Ф时～ 无意义。 :11 A,F1 例1. 平庸空间 设X是一个集合，令T ={X ,Φ}，容易验证T是X的一个拓扑，称为X的平庸拓扑，称 (X，T )为平庸空间。在平庸空间中，有且只有两个开集:X ,Φ;有且只有两个开集:X ,Φ。 例2.离散空间 设X是一个集合，令T =P (X)，易知T是X的一个拓扑，称为X的离散拓扑，称(X，T )为离散空间。在离散空间中，每一个子集都是开集，每一个子集都是开集。离散空间可以记作(X，P (X)) 。 例3. 设X={a,b,c},令T ={Φ ,{a} ,{a,b},X },可以验证T是X的一个7 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 拓扑，因此(X，T )为一个拓扑空间。它既不是平庸拓扑，又不是离散拓扑。 说明: 对X={a,b,c}～可以为其构造出29个拓扑～其中平庸拓扑最小～离散拓扑最大。可见对同一个集合～它可以有不同的拓扑。 例4.有限补拓扑空间 设X是一个集合，令T ={U X | U' 是X的一个有限子集 }?{Φ}。 , 易验证T是X的一个拓扑，称其为X的有限补拓扑，(X，T )称为有限补拓扑空间。下面验证T 满足拓扑定义中的?成立 A,,, 设T T ，若T= Φ，则 T ;若存在A?Ф, A?T ,,1 1 1 00 :A,T1 ,A,,,则是X的有限子集，所以 T 。所以?成立。 AAA),,:(::A,T0A,T11 A,T 1 问题:当X是一个有限集合时～X 的有限补拓扑空间又是已知的什么拓扑空间 , 例5. 可数补拓扑空间 设X是一个集合，令T ={U X | U' 是X的一个可数子集 }?{Φ}。 , 易验证T是X的一个拓扑，称其为X的可数补拓扑，(X，T )称为可数补拓扑空间。(课下验证) 问题:当X是一个可数集合时～X 的可数补拓扑空间又叫做什么拓扑空间 ,,离散拓扑空间, 。当X是有限时～与什么空间是同一个空间,,有限拓扑空间, 三、邻域与邻域系、聚点、导集，闭集，闭包 1. 邻域邻域系的定义 , 定义1.1.3 设(X,T )是一个拓扑空间，x?X,U X,如果存在开集V ?T使得x?V U ，则称U是x的一个邻域。点x的所有邻域构成的集, 族称为点x的邻域系。 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 由定义，若U是包含x的开集，那么它一定是x的一个邻域，称U是点x的一个开邻域。 说明:由于X的子集A是X作为度量空间的开集与A是X作为拓扑空间的开集是一回事，所以包含x的集合U是X作为度量空间x的邻域U是X作,为拓扑空间x的邻域。 定理1.1.2 X是一个拓扑空间， UX， 则U是X的一个开集U是其,, 内每一点的邻域。 证明:“”显然。 , “” 若U =Φ，则结论成立。若U ?Φ，由条件对每一个x?U，存在开, 集V 使x?VU ，因此，所以 为开集。 ,UxVU,,,::{}UV,:xxxx,,xU,xUxU ,推论 U是X的一个开集 U可以示为开邻域之并。 2. 导集，闭集，闭包的概念 ,定义1.1.4 设X是拓扑空间，AX，x?X，如果对x的每一个邻域U都有U?(A-{x})?Φ,则称点x是集合A的一个聚点。 集合A的所有聚点构成的集合称为A的导集，记作d(A)。如果x?A，并且x不是A的凝聚点，既存在x的一个邻域U使得U?(A-{x})=Φ，则称点x是集合A的一个孤立点。 ,AAA集合A与A的导集d(A)的并A?d(A)称为集合A的闭包，记作或。即= A?d(A) 。 说明 (1)A的孤立点一定的属于A，但A的极限点不一定属于A; (2)凝聚点、孤立点、导集都是相对于X的某个拓扑而言的，它与拓扑有关。 因此在谈这些问题时一般都需要明确是相对于那个拓扑来说的。同时也可知: (3) 欧氏空间中有关这几个概念的结论在一般拓扑空间中不见得的成立。 ,(4)若Ad(A)～则称 A为自密集～若A=d(A)～则称A为完全集。若d(A)?A=Φ～则称A为孤立点集。 (5)在离散空间中～由于d(A)=Φ～既没有任何极限点～所以任何子集都是闭集。,我们已知任何子集是开集,。而在平庸空间中～d({x。})=X-{x。}～若Acoupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, a9 part from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 多于一个点～则d(A)=X～所以在平庸空间中任何真子集都不是闭集 3. 导集，闭集，闭包的性质 定理1.1.3 设X是一个拓扑空间AX，则 , ?d(Φ)=Φ ? 若AB，则d(A)d(B) ,, ?d(A?B) =d(A)?d(B) ?d(d(A))A?d(A) , 证明 ?由于对于每一点x?X和点x的任何一个邻域U有U?(Φ-{x})=Φ,所以xd(Φ),因此d(Φ)=Φ 。 , ? 如果x?d(A),U是x一个邻域,由于U?(A-{x})?Φ,所以U?(B-{x}) ?Φ，因此x?d(B)。这证明了d(A) d(B) 。 , ? 据?及A,B, A?B得知d(A)，d(B) , d(A?B)，所以d(A)?d(B) , d(A?B)，下证d(A?B) , d(A)?d(B) 。设x?d(A?B)，则对x的任何一个邻域U有U?(A?B -{x})?Φ，即U?[(A-{x})?(B-{x})]= [U?(A-{x})]?[U?(B-{x})]?Φ，所以U?(A-{x})?Φ或U?(B-{x})?Φ，所以x?d(A)或x?d(B)，所以x?d(A)?d(B)，所以d(A?B) = d(A)?d(B) 。 ? 设x A?d(A)，则 xA 且 xd(A) ，所以存在x的一个邻域U 使U,,, ,?(A-{x})=Φ ，任意选取x的一个开邻域V，使得VU ，这是我们也有V?(A-{x})=Φ ,由于xA ，所以V?A =Φ，这也就是说，V中的任何一个点都不, 是A中的点，因此对于任何y?V，有V?(A-{y})=Φ,由于V是y的一个邻域，因此y不是A的凝聚点，即yd(A) 。这说明V中没有A的任何一个凝聚点。于, 是x有一个邻域V与A的导集d(A) 无交，即V?d(A)=Φ，所以V?(d(A)-{x})=Φ，所以 xd(d(A)) 。将以上给出的论证概括起来便是:只要x A?d(A)，,, ,便有xd(d(A))，这就是说d(d(A))A?d(A) 。证毕。 , 注:d(d(A)) d(A) ， d(A) d(d(A)) ,, 。 ,,定理1.1.4 设X是一个拓扑空间，AX，则A是闭集d(A)A。 , 证明 “”设A是闭集，则A'是开集，如果xA，则x?A'，则A'是x的,, ,一个邻域，它满足条件:A?A'=Ф，因此xd(A)。于是我们有d(A)A。 , coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 “”设d(A)A。如果x?A' ，则xA，所以xd(A)，由聚点的定义x,,,, 有一个邻域U使U?(A-{x})=Φ，从而U?A=Φ，也即U A'，这证明，对于任,何x?A'，A'是x的一个邻域，因此A'是开集。 定理1.1.5 拓扑空间X的子集A是闭集 。 AA,, 证 A为闭集 d(A) A A?d(A)=A,即 。 AA,,,, 定理1.1.6 X是拓扑空间，对于任意的集合A,BX ，有 , ABAB::,? ; ? ; ? ; ? AA,AA,,,, 证明 „ „用到 d(A?B) =d(A)?d(B) 和d(d(A))A?d(A) , 定理1.1.7 拓扑空间X的任何一个子集A的闭包都是闭集。 定理1.1.8 设X是拓扑空间，F 是由空间X中所有的闭集构成的族，则对 AB,:于X的每个子集A，有 。即集合A的闭包等于包含A的所有闭BBA,,F, 集之交。 :B证明 由于A包含于 ，然而后者是一闭集，所以AB,:;BBA,,F,BBA,,F, AB,:AAA,另一方面，因为是闭集，并且 ，所以:BA, ，所以 。 BBA,,F,BBA,,F, A说明 因是包含A的闭集，而由定理，又包含于任何一个包含A的闭集之中。因此我们有结论:一个集合的闭包是包含着这个集的最小闭集。 ,定理1.1.4 设X是一个拓扑空间，AX，则 xA,,,,,UU,有UA:,? ;?; 若，则ABABdAdB,,,(()())？x ABAB::,? ，由?可得。 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, a11 part from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 一个令人关心的问题是，是否拓扑空间真的要比度量空间的范围更广一点, 换句话说，是否每一个度量空间都可以由某一个度量诱导出来, 定义2.2.3 设(X，T )是拓扑空间。如果存在X的一个度量ρ使得拓扑T就是由ρ诱导出来的拓扑T ，则称(X，T )是一个可度量化空间。 ρ 注: ?是否每一个拓扑空间都是可度量化空间, 回答是否定的。因为由?2.1习题2可知，每一个只含有限点的度量空间作为拓扑空间都是离散空间。然而一个平庸空间如果含有多于一个点的话，它肯定不是离散空间，因此它不是可度量的。例2.2.3给出的拓扑空间含三个点，但不是离散空间，就不是可度量化的; ? 由此看来，拓扑空间确实比度量空间范围更广; ? 拓扑空间在什么条件下可度量化,后面将由专门讨论。 二 拓扑空间之间的连续映射及同胚 1. 拓扑空间之间的连续映射及性质 定义2.2.4 设X、Y是两个拓扑空间，f:X?Y ,如果Y中每个开集U的原 -1像f(U)是X中的一个开集，则称f是从X到Y的一个连续映射，或简称映射f连续。 问题:常值映射连续吗,——连续。可见连续映射不一定是一一映射。 注 设X、Y是两个度量空间，f:X?Y连续。由于视X,Y为拓扑空间时，其开集与X,Y作为度量空间时的开集一样，所以由该定义和Th2.1.4知，X,Y都作为拓扑空间时，f:X?Y也连续。可见拓扑空间的连续是度量空间之间连续的推广。 定理2.2.1 设X、Y、Z是拓扑空间，则 ? 恒同映射i:X?X是一个连续映射; X ? 如果f:X?Y 连续，g:Y?Z连续，则g。f:X?Z也连续。 ,1证明 ?如果U是X的一个开集，则iU()=U,当然也是X的开集，所以i XX 连续。 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 -1? 设f:X?Y 连续，g:Y?Z连续，设W是Z的开集,由于g 连续，所以g(W) -1-1-1是Y中开集;又因为f连续，所以f[g(W)]是X中的开集。因此(g。f)(W)-1-1=f[g(W)]是X中的开集。这证明g。f连续。 2. 拓扑空间之间的同胚及性质 在数学的许多学科中都涉及两类基本对象。例如在线性代数中我们考虑线性空间和线性变换，在群论中我们考虑群和同态，在集合论中我们考虑集合与映射，在不同的几何学中考虑各自的图形和各自的变换等等。并且对于后者都要提出一类予以重视，例如线性代数中的(线性)同构，群论中的同构，集合论中的一一映射，以及初等几何中的刚体运动(即平移加旋转)等等，我们现在已经提出了两类基本对象，即拓扑空间和连续映射。下面将从连续映射中挑出一类予以关注。这就是同胚映射。 定义2.2.5 设X和Y都是拓扑空间，如果f:X?Y是一个一一映射，并且f -1和f 都连续，则称f是一个同胚映射或同胚。 -1注:? 直观上说～f连续表示不撕裂～ f 连续表示不粘连。f:X?Y是一个同胚～表示X到Y不撕裂、不粘连。 映射f是一一映射～一定连续吗, ? 不见得连续。如～设f:Q?Z 是一一映射,因为有从Q?Z的单射～也有从++ Z?Q的单射～据定理1.7.9有这样的一一映射f, + 若取Q的拓扑为平庸拓扑～Z的拓扑为离散的拓扑～则f不连续。 + ? 连续的一一映射一定同胚吗, nn不一定。如～令X=R ,取它的拓扑为离散拓扑～Y= R ,取它的拓扑为通常 ,1iiXY:,度量诱导的拓扑。映射是连续的、一一的～但不连续。nnRR ,,11？aXaiaa,,,{})(){}为开集，但(iXY:,在Y中是闭集。所以不是同胚。 nnRR 定理2.2.2 设X、Y、Z是拓扑空间，则 ? 恒同映射i:X?X是一个同胚; X -1? 如果f:X?Y是一个同胚，则f: Y?X也是一个同胚; ? 如果f:X?Y 和g:Y?Z都是同胚，则g。f:X?Z也是一个同胚。 证明:(以下证明中的根据，可见定理2.2.1，定理1.5.3，定理1.5.4) coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, a13 part from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, -1? 恒同映射i 是一个一一映射，并且i =i 都是连续的，从而i 是一 XXXX个同胚。 -1 ? 设f:X?Y是同胚，因此f是一个一一映射，并且f和f都是连续的。 -1-1-1-1-1于是f也是一个一一映射并且f和(f)=f也都连续，所以f: Y?X也是一个同胚。 ? 如果f:X?Y 和g:Y?Z都是同胚，因此f和g是一个一一映射,并且f -1,-1-1和f，g和g 都是连续的，因此g。f也是一一映射，并且g。f和(g。f) = -1-1f 。g 都是连续的，所以g。f:X?Z也是一个同胚。 定义2.2.6 设X、Y是拓扑空间，如果存在一个同胚f:X?Y，则称拓扑空间X与拓扑空间Y是同胚的，或称X与Y同胚，或称X同胚与Y 。 定理2.2.3设X、Y、Z是拓扑空间，则 ? X与X同胚; ? 若X与Y同胚,则Y与X也同胚; ? 若X与Y同胚,则Y与Z同胚,则X与Z也同胚。 证明:从定理2.2.2直接可得。 说明:? 在拓扑空间组成的族中，同胚关系是一个等价关系。因此同胚关 系将拓扑空间族分成互不相交的等价类，使得属于同一类的拓扑空间彼此同胚，属于不同类的拓扑空间彼此不同胚。 ? 拓扑空间的某种性质P，如果为某一拓扑空间所具有，则与其同胚的拓扑空间也具有，则称性质P是一个拓扑不变性质或拓扑不变量。简言之，拓扑不变性是同胚的拓扑空间都具有的性质。例如后面将要讨论的集合为开集、闭集、点集的闭包与导集、点的邻域、序列的收敛性以及拓扑空间的连通性、紧致性等都是拓扑不变性质。拓扑不变性质简称拓扑性质。 ? 拓扑学的中心任务就是研究拓扑空间的拓扑不变性质。 研究拓扑空间的拓扑性质很有意义。如果我们研究某一问题时，研究的是这coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 吉首大学数学与统计学院 点集拓扑教案 一空间的拓扑性质，我们可以转化为对其同胚空间的研究，这样就有可能使某些难处理的问题变得简单。 当然，证明两个空间的同胚有时是非常困难的事情。实际上我们常常利用拓扑性质来区分空间，说明它们不同胚。一个空间有拓扑性质P，而另一个空间没有性质P，则这两个空间就一定不是同胚的。 至此我们已经做完了将数学分析中的欧氏空间和欧氏空间之间的连续函数的概念，经由度量和度量空间之间的连续映射，一直抽象为拓扑空间和拓扑空间之间的连续映射这样一个在数学史上经过了很长时间才完成的工作。 在数学的发展过程中，对所研究的问题不断地加以抽象这种做法屡见不鲜的，但每一次的抽象都是把握住旧的研究对象(或其中的某个方面)的精髓而进行的一次提升，是一个去粗取精的过程。也正因为如此，新的概念和理论往往有更多的包容。拓扑学无疑也是如此，一方面它使我们对“空间”和“连续”有了更为纯正的认识，另一方面也包含了无法列入以往理论中的新的研究对象(特别是许多无法作为度量处理的映射空间)。这一些我们在学习过程中必然会不断地加深体会。 为了对新旧概念的区别有更加深刻的印象，在这两节中给出了一些例子。客观地讲这些例子除去欧氏空间(包括实数空间和Hilbert空间)其他都显得有点怪，明显的是为澄清概念而构造出来的。这些例子只是帮助我们更好地掌握拓扑学的工具。不要误认为拓扑学就是数学分析中的连续函数再加上某些不常见的例子。 补充命题 ，见余玄冰编译《点集拓扑》P 。 50-52 命题1 设X,Y都是拓扑空间，f:X?Y和g:Y?X都连续，且g?f = i, f?g X -1= i，则f是同胚，而且实际上g=f . Y 命题2 映射f:X?R连续??b?R，{ x | f(x) b }都是开集. α α α 命题3 令f , g:X?R连续. 则 ?|f|(α>0)是连续的.(|f|= |f(x)|) ? a f + b g 连续，其中a,b?R; ? f ?g是连续的; 1? 若在X上，f(x)?0，则是连续的. f 15 coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave, 注: 后面证明定理4.2.5 [ Borsuk—Ulam定理 ]时用到命题3 . coupling Centre shall be checked, the second poured concrete strength of more than 70%. In the process of reaming or boring, apart from the special provisions in factory, does not allow any work that may affect the coupling Centre, such as irrigation, to cylinder connections to flood the condenser pipes and so on. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Mounting bolts should be added lubricants, typing gently with a small hammer, not too loose or too tight, packed the bolts with the corresponding seal screw hole should be marked. 6.4.5.4.2 coupling reamed and pairings is close to the diameter of bolt bearing good hinge holes, dressed with a good two coupling bolts before you can move the rotor, boring-reaming hole in turn. Coupling bolts must be tight coupling is symmetrical diameter, in boring, reaming, two bolts out. Well boring, hinge bolt hole flange with coupling of vertical, not oblique. Due to the coupling bolts in accordance with hinge holes corresponding to the diameter of the screw holes one by one after processing, processing back bolt installation location cannot be changed, so according to bolt after the actual weight difference of nut and lock washer to match. General requirements for coupling two symmetrical diameter bolt, nut and lock washer's total weight difference should not be greater than 10 g and knocks stamped mark. (See Figure 18) The coupling bolts apply lubricant, and then gently with copper home, not too loose or too tight. Bolt Assembly is finished, symmetrical tightening all bolts to 30%, 50%, 70%, 100% force, and check the coupling flutter and before connecting the circle gradually changes its value should be less than 0.02mm. Tightening torque of the coupling bolts shall comply with the technical requirements manufacturers drawings, with a dynamometric wrench or log tension tools. Three wheel rotor swing test coupling connections should be checked: taking leave,