топология / Tom Dieck T. Algebraic topology (EMS, 2008)

8 Chapter 1. Topological Spaces

Since p and q are quotient maps, ˛ is a homeomorphism. The continuous periodic functions f W R ! R; f .x C 1/ D f .x/ therefore correspond to continuous maps R=Z ! R and to

continuous maps S1 ! R via composition with q or p. In a similar manner one obtains a homeomorphism C=Z Š C .

5. Let f W A ! B and g W B ! C be continuous. If f and g are quotient maps, then gf is a quotient map. If gf is a quotient map, then g is a quotient map. If gf D id, then g is a quotient map.

1.3 Products and Sums

Let ..Xj ; Oj / j j 2 J / be a family of topological spaces. The product set X D

category TOP. Note that for infinite J , open sets in the product are quite large; a

projections prj are continuous. A set map f W Y ! X from a space Y into X is

The product of X1; X2 is denoted X1 X2, and we use f1 f2 for the product of maps. The “identity” id W X1 .X2 X3/ ! .X1 X2/ X3 is a homeomorphism. In general, the topological product is associative, i.e., compatible with arbitrary bracketing. The canonical identification Rk Rl D RkCl is a homeomorphism.

1.3.2 Pullback. Let f W X ! B and g W Y ! B be continuous maps. Let Z D f.x; y/ 2 X Y j f .x/ D g.y/g with the subspace topology of X Y . We have the projections onto the factors F W Z ! Y and G W Z ! X. The commutative diagram

Z F Y

Gg

f

XB

is a pullback in TOP. The space Z is sometimes written Z D X B Y and called the product of X and Y over B (the product in the category TOPB of spaces over B).

10 Chapter 1. Topological Spaces

and gjk ı gij D gik holds, considered as maps from Uij \ Uik to Ukj \ Uki . Given a clutching datum, we have an equivalence relation on the disjoint sum

qj 2J Uj :

x 2 Ui y 2 Uj , x 2 Uij and gij .x/ D y:

Let X denote the set of equivalence classes and let hi W Ui ! X be the map which

(1.3.7) Proposition. Let .Ui ; Uij ; gij / be a clutching datum. Assume that the Ui

are topological spaces, the Uij Ui open subsets, and the gij W Uij ! Uji homeomorphisms. Let X carry the quotient topology with respect to the quotient map p W qj 2J Uj ! X. Then the following holds:

(1)The map hi is a homeomorphism onto an open subset of X and p is open.

(2)Suppose the Ui are Hausdorff spaces. Then X is a Hausdorff space if and

1.3.8 Euclidean space with two origins. The simplest case is obtained from

Then the graph of ' in R R is not closed. The resulting locally Euclidean space

the colimit topology with respect to this family if one of the equivalent statements hold:

(2)C is closed in X if and only if Xj \ C is closed in Xj for each j .

(3)A set map f W X ! Z into a space Z is continuous if and only if the restrictions f jXj W Xj ! Z are continuous.

(1.3.9) Example. Let X be a set which is covered by a family .Xj j j 2 J / of subsets. Suppose each Xj carries a topology such that the subspace topologies of

Xi \ Xj in Xi and Xj coincide and these subspaces are closed. Then there is a unique topology on X which induces on Xj the given topology. The space X has the colimit topology with respect to the Xj . Þ

Problems

X X. Let f; g W X ! Y be continuous maps into a Hausdorff space. Then the coincidence set A D fx j f .x/ D g.x/g is closed in X. Hint: Use (1.3.3).

4. A discrete space is the topological sum of its points. There is always a canonical homeomorphism X qj Yj Š qj .X Yj /. For each y 2 Y the map X ! X Y , x 7!.x; y/ is an embedding. If f W X ! Y is continuous, then W X ! X Y , x 7!.x; f .x// is an embedding. If Y is a Hausdorff space, then is closed.

1.4 Compact Spaces

A family A D .Aj j j 2 J / of subsets of X is a covering of X if X is the union of the Aj . A covering B D .Bk j k 2 K/ of X is a refinement of A if for each k 2 K

there exists j 2 J such that Bk Aj . If X is a topological space, a covering A D .Aj j j 2 J / is called open (closed) if each Aj is open (closed). A covering B D .Bk j k 2 K/ is a subcovering of A if K J and Bk D Ak for k 2 K. We say B is finite or countable if K is finite or countable. A covering A is locally

finite if each x 2 U has a neighbourhood U such that U \ Aj ¤ ; only for a finite number of j 2 J . It is called point-finite if each x 2 X is contained only in a finite

number of Aj .

A space X is compact if each open covering has a finite subcovering. (In some texts this property is called quasi-compact.) By passage to complements we see:

If X is compact, then any family of closed sets with empty intersection contains a finite family with empty intersection. A set A in a space X is relatively compact

if its closure is compact. We recall from calculus the fundamental Heine–Borel Theorem: The unit interval I D Œ0; 1 is compact.

A space X is compact if and only if each net in X has a convergent subnet (an accumulation value). A discrete closed set in a compact space is finite. Let X be compact, A X closed and f W X ! Y continuous; then A and f .X/ are compact.

(1.4.1) Proposition. Let B, C be compact subsets of spaces X, Y , respectively. Let U be a family of open subsets of X Y which cover B C . Then there exist

12 Chapter 1. Topological Spaces

open neighbourhoods U of B and V of C such that U V is covered by a finite subfamily of U. In particular the product of two compact spaces is compact.

One can show that an arbitrary product of compact spaces is compact (Theorem of Tychonoff ).

(1.4.2) Proposition. Let B and C be disjoint compact subsets of a Hausdorff space X. Then B and C have disjoint open neighbourhoods. A compact Hausdorff space is normal. A compact subset C of a Hausdorff space X is closed.

(1.4.3) Proposition. A continuous map f W X ! Y from a compact space into a Hausdorff space is closed. If, moreover, f is injective (bijective), then f is an embedding (homeomorphism). If f is surjective, then it is a quotient map.

(1.4.4) Proposition. Let X be a compact Hausdorff space and f W X ! Y a quotient map. The following assertions are equivalent:

(1)Y is a Hausdorff space.

(2)f is closed.

(3) R D f.x1; x2/ j f .x1/ D f .x2/g is closed in X X.

Let X be a union of subspaces X1 X2 . Recall that X carries the colimit-topology with respect to the filtration .Xi / if A X is open (closed) if

and only if each intersection A \ Xn is open (closed) in Xn. We then call X the colimit of the ascending sequence .Xi /. (This is a colimit in the categorical sense.)

(1.4.5) Proposition. Suppose X is the colimit of the sequence X1 X2 . Suppose points in Xi are closed. Then each compact subset K of X is contained in some Xk .

A space is locally compact if each neighbourhood of a point x contains a compact neighbourhood. An open subset of a locally compact space is again locally compact.

Let X be a Hausdorff space and assume that each point has a compact neighbourhood. Let U be a neighbourhood of x and K a compact neighbourhood. Since K is normal, K \ U contains a closed neighbourhood L of x in K. Then L is compact and a neighbourhood of x in X. Therefore X is locally compact. In particular, a compact Hausdorff space is locally compact. If X and Y are locally compact, then X Y is locally compact.

Let X be a topological space. An embedding f W X ! Y is a compactification of X if Y is compact and f .X/ dense in Y .

The following theorem yields a compactification by a single point. It is called the

Alexandroff compactification or the one-point compactification. The additional point is the point at infinity. In a general compactification f W X ! Y , one calls

the points in Y X f .X/ the points at infinity.

(1.4.6) Theorem. Let X be a locally compact Hausdorff space. Up to homeomorphism, there exists a unique compactification f W X ! Y by a compact Hausdorff space such that Y X f .X/ consists of a single point.

(1.4.8) Theorem. Let the locally compact Hausdorff space M ¤ ; be a union of closed subsets Mn, n 2 N. Then at least one of the Mn contains an interior point.

A subset H of a space G is called locally closed, if each x 2 H has a neighbourhood Vx in G such that H \ Vx is closed in G.

(1.4.9) Proposition. .1/ Let A be locally closed in X. Then A D U \ C with U open and C closed. Conversely, if X is regular, then an intersection U \ C , U open, C closed, is locally closed.

Problems

1. Dn=Sn 1 is homeomorphic to Sn. For the proof verify that

q

Dn ! Sn; x 7! 2 1 kxk2x; 2kxk2 1

induces a bijection Dn=Sn 1 ! Sn.

2. Let f W X C ! R be continuous. Assume that C is compact and set g.x/ D supff .x; c/ j c 2 C g. Then g W X ! R is continuous.

3. Let X be the colimit of an ascending sequence of spaces X1 X2 . Then the Xi are subspaces of X. If Xi XiC1 is always closed, then the Xj are closed in X.

be the vector space of all sequences .x1; x2; : : : / of real numbers which are eventually zero. Let Rn be the subspace of sequences with xj D 0 for j > n. Give R1 the colimit topology with respect to the subspaces Rn. Then addition of vectors is a continuous map R1 R1 ! R1. Scalar multiplication is a continuous map R R1 ! R1. (Thus R1 is a topological vector space.) A neighbourhood basis of zero consists of the intersec-

dimensional linear subspaces. One can also consider the metric topology with respect to the metric d..xi /; .yi // D Pi .xi yi /2 1=2; denote it by R1d . The identity R1 ! R1d is

continuous. The space R1 is separated.

14 Chapter 1. Topological Spaces

1.5 Proper Maps

A continuous map f W X ! Y is called proper if it is closed and the pre-images f 1.y/; y 2 Y are compact.

(1.5.1) Proposition. Let K be compact. Then pr W X K ! X is proper. If f W X ! Y is proper and K Y compact, then f 1.K/ is compact. Let f and g be proper; then f g is proper.

As a generalization of the theorem of Tychonoff one can show that an arbitrary product of proper maps is proper.

(1.5.2) Proposition. Let f W X ! X0 and g W X0 ! X00 be continuous.

(1)If f and g are proper, then g ı f is proper.

(2)If g ı f is proper and f surjective, then g is proper.

(1.5.3) Proposition. Let f W X ! Y be injective. Then the following are equivalent:

(1)f is proper.

(2)f is closed.

(1.5.5) Proposition. Let f be a continuous map of a Hausdorff space X into a locally compact Hausdorff space Y . Then f is proper if and only if each compact set K Y has a compact pre-image. If f is proper, then X is locally compact.

(1.5.6) Proposition. Let f W X ! X0 and g W X0 ! X00 be continuous and assume that gf is proper. If X0 is a Hausdorff space, then f is proper.

(1.5.7) Theorem. A continuous map f W X ! Y is proper if and only if for each space T the product f id W X T ! Y T is closed.

Problems

1. A map f W X ! Y is proper if and only if the following holds: For each net .xj / in X and each accumulation value y of .f .xj // there exists an accumulation value x of .xj / such

that f .x/ D y.

2.Let X and Y be locally compact Hausdorff spaces, let f W X ! Y be continuous and f C W XC ! Y C the extension to the one-point compactification. Then f C is continuous,

if f is proper.

3.The restriction of a proper map to a closed subset is proper.

4.Let f W X ! Y be proper and X a Hausdorff space. Then the subspace f .X/ of Y is a Hausdorff space.

5.Let f W X ! Y be continuous. Let R be the equivalence relation on X induced by f , and denote by p W X ! X=R the quotient map, by h W X=R ! f .X/ the canonical bijection,

and let i W f .X/ Y . Then f D i ı h ı p is the canonical decomposition of f . The map

fis proper if and only if p is proper, h a homeomorphism, and f .X/ Y closed.

1.6Paracompact Spaces

x inclusion Bj Uj .

A point-finite open covering of a normal space has a shrinking.

A space X is called paracompact if it is a Hausdorff space and if each open covering has an open, locally finite refinement. A closed subset of a paracompact space is paracompact. A compact space is paracompact.

A paracompact space is normal. Suppose the locally compact Hausdorff space X is a countable union of compact sets. Then X is paracompact. Let X be paracompact and K be compact Hausdorff. Then X K is paracompact. A metric space is paracompact.

1.7 Topological Groups

A topological group .G; m; O/ consists of a group .G; m/ with multiplication m W G G ! G, .g; h/ 7!m.g; h/ D gh and a topology O on G such that the multiplication m and the inverse W G ! G, g 7!g 1 are continuous. We de-

note a topological group .G; m; O/ usually just by the letter G. The neutral element will be denoted by e (also 1 is in use and 0 for abelian groups). The left translation

lg W G ! G, x 7!gx by g 2 G in a topological group is continuous, and the rules lg lh D lgh and le D id show it to be a homeomorphism. For subsets A and B of a group G we use notations like aB D fab j b 2 Bg, AB D fab j a 2 A; b 2 Bg, A2 D AA, A 1 j a 2 Ag, and similar ones.

A group G together with the discrete topology on the set G is a topological group, called a discrete (topological) group.

The additive groups of the real numbers R, complex numbers C, and quaternions H with their ordinary topology are topological groups, similarly the multiplicative

16 Chapter 1. Topological Spaces

groups R , C and H of the non-zero elements. The multiplicative group RC of the positive real numbers is an open subgroup of R and a topological group. The complex numbers of norm 1 are a compact topological group S1 with respect to multiplication. The exponential function expW R ! RC is a continuous homomorphism with the logarithm function as a continuous inverse. The complex exponential function expW C ! C is a surjective homomorphism with kernel f2 i n j n 2 Zg, a discrete subgroup of C.

The main examples of topological groups are matrix groups. In the vector space Mn.R/ of real .n; n/-matrices let GLn.R/ be the subspace of the invertible matrices. Since the determinant is a continuous map, this is an open subspace.

Matrix multiplication and passage to the inverse are continuous, since they are given by rational functions in the matrix entries. This makes the general linear group GLn.R/ into a topological group. Similarly for GLn.C/. The determinant is a continuous homomorphism det W GLn.R/ ! R with kernel the special linear group SLn.R/; similarly in the complex case.

Let O.n/ D fA 2 Mn.R/ j At A D Eg be the group of orthogonal .n; n/-

matrices (At transpose of A; E unit matrix). The set O.n/ is a compact subset in Mn.R/. Hence O.n/ is a compact topological group (the orthogonal group). The

of unitary .n; n/-matrices is a compact topological group (unitary group). The topological groups SO.2/, U.1/, and S1 are isomorphic. The special unitary group

SU.n/ is the compact subgroup of U.n/ of matrices with determinant 1. The multiplicative group of quaternions of norm 1 provides S3 with the structure of a topological group. This group is isomorphic to SU.2/. From linear algebra one knows about a surjective homomorphism SU.2/ ! SO.3/ with kernel ˙E (a twofold covering); for this and other related facts see the nice discussion in [27, Kapitel IX]. For more information about matrix groups, also from the viewpoint of manifolds and Lie groups, see [29]; there you can find, among others, the symplectic groups Sp.n/ and the Spinor groups Spin.n/. The isomorphisms SU.2/ Š Spin.3/ Š Sp.1/ hold, and these spaces are homeomorphic to S3.

If G and H are topological groups, then the direct product G H with the

product topology is a topological group. The n-fold product S1 S1 is called an n-dimensional torus.

The trivial subgroup is often denoted by 1 (in a multiplicative notation) or by 0 (in an additive notation). The neutral element will also be denoted 1 or 0. The symbol H C G is used for a normal subgroup H of G. The notation H K or H G K means that H and K are conjugate subgroups of G.

A homomorphism f W G ! H between topological groups is continuous if it is continuous at the neutral element e.

If G is a topological group and H G a subgroup, then H , with the subspace topology, is a topological group (called a topological subgroup). If H G is a

x subgroup, then the closure of H is also. If H is a normal subgroup, then H is also.

1.8 Transformation Groups

A left action of a topological group G on a topological space X is a continuous

map W G X ! X, .g; x/ 7!gx such that g.hx/ D .gh/x and ex D x for g; h 2 G, e 2 G the unit, and x 2 X. A (left) G -space .X; / consists of a space

X and a left action of G on X. The homeomorphism lg W X ! X, x 7!gx is called left translation by g. We also use right actions X G ! X, .x; g/ 7!xg;

they satisfy .xh/g D x.hg/ and xe D x. For A X and K G we let KA D fka j k 2 K; a 2 Ag. An action is effective if gx D x for all x 2 X implies g D e. The trivial action has gx D x for g 2 G and x 2 X.

The set R D f.x; gx/ j x 2 X; g 2 Gg is an equivalence relation on X. The set of equivalence classes X mod R is denoted by X=G. The quotient map

q W X ! X=G is used to provide X=G with the quotient topology. The resulting space X=G is called the orbit space of the G-space X. A more systematic notation

for the orbit space of a left action would be GnX. The equivalence class of x 2 X is the orbit Gx through x. An action is transitive if it consists of a single orbit. The set Gx D fg 2 G j gx D xg is a subgroup of G, the isotropy group or the stabilizer of the G-space X at x. An action is free if all isotropy groups are

trivial. We have Ggx D gGx g 1. Therefore the set Iso.X/ of isotropy groups of X

consists of complete conjugacy classes of subgroups. If it contains a finite number of conjugacy classes, we say X has finite orbit type.

A subset A of a G-space is called G -stable or G -invariant if g 2 G and a 2 A implies ga 2 A. A G-stable subset A is also called a G -subspace. For each subgroup H of G there is an H -fixed point set of X,

XH D fx 2 X j hx D x for all h 2 H g:

Suppose X and Y are G-spaces. A map f W X ! Y is called a G-map or a G -equivariant map if for g 2 G and x 2 X the relation f .gx/ D gf .x/

holds. In general, the term “equivariant” refers to something related to a group action. Left G-spaces and G-equivariant maps form the category G- TOP. This

map f =G W X=G ! Y =G. We have the notion of an equivariant homotopy or G -homotopy Ht : this is a homotopy such that each Ht is a G-map.

(1.8.1) Proposition. Let X be a G-space, A G and B X. If B is open then AB is open. The orbit map p W X ! X=G is open.