nLab
compact-open topology

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Mapping space

Contents

Idea

A natural topology on mapping spaces of continuous functions, important because of its role in exhibiting locally compact topological spaces to be exponentiable, as demonstrated below, culminating in Corollary 1.

The compact-open topology on the set of continuous functions XYX \to Y is generated by the subbasis of subsets U KC(X,Y)U^K \subset C(X,Y) that map a given compact subspace KXK \subset X to a given open subset UYU \subset Y, whence the name.

When restricting to continuous functions between compactly generated topological spaces one usually modifies this definition to a subbase of open subsets U ϕ(K)U^{\phi(K)}, where now ϕ(K)\phi(K) is the image of a compact topological space under any continuous function ϕ:KX\phi \colon K \to X. This definition gives a cartesian internal hom in the category of compactly generated topological spaces (see also at convenient category of topological spaces).

The two definitions agree when the domain XX is a compactly generated Hausdorff space, but not in general. Beware that texts on compactly generated spaces nevertheless commonly say “compact-open topology” for the second definition.

Definition

Notation

Let (X,𝒪 X)(X, \mathcal{O}_{X}) and (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be topological spaces. We denote by Y XY^{X} the set of continuous maps from (X,𝒪 X)(X,\mathcal{O}_{X}) to (Y,𝒪 Y)(Y, \mathcal{O}_{Y}).

Remark

Other common notations for Y XY^{X} are Map(X,Y)Map(X,Y) or C(X,Y)C(X,Y).

Notation

Let (X,𝒪 X)(X, \mathcal{O}_{X}) be a topological space. We denote by 𝒪 X c\mathcal{O}^{c}_{X} the set of subsets of XX which are compact with respect to 𝒪 X\mathcal{O}_{X}.

Notation

Let (X,𝒪 X)(X, \mathcal{O}_{X}) be a topological space. Let AA be a subset of XX. We denote by A¯\overline{A} the topological closure of AA in XX with respect to 𝒪 X\mathcal{O}_{X}.

Definition

A topological space (X,𝒪 X)(X, \mathcal{O}_{X}) is locally compact if the set of compact neighbourhoods of points of XX form a neighbourhood basis of XX. That is to say: for every xXx \in X and every neighbourhood NN of xx, there is a V𝒪 X cV \in \mathcal{O}^{c}_{X} and a U𝒪 XU \in \mathcal{O}_{X} such that xUVNx \in U \subset V \subset N.

Remark

There are many variations on this definition, which can be found at locally compact space. These are all equivalent if (X,𝒪 X)(X, \mathcal{O}_{X}) is Hausdorff. We do not however make the assumption that (X,𝒪 X)(X, \mathcal{O}_{X}) is Hausdorff.

Notation

Let (X,𝒪 X)(X, \mathcal{O}_{X}) and (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be topological spaces. Given A𝒪 X cA \in \mathcal{O}^{c}_{X} and U𝒪 YU \in \mathcal{O}_{Y}, we denote by M A,UM_{A,U} the set of continuous maps f:XYf : X \rightarrow Y such that f(A)Uf(A) \subset U.

Definition

Let (X,𝒪 X)(X, \mathcal{O}_{X}) and (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be topological spaces. The compact-open topology on Y XY^{X} is that with sub-basis given by the set of sets M A,UM_{A,U} such that A𝒪 X cA \in \mathcal{O}^{c}_{X} and U𝒪 YU \in \mathcal{O}_{Y}.

Notation

Let (X,𝒪 X)(X, \mathcal{O}_{X}) and (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be topological spaces. We shall denote the compact-open topology (def. 2) on Y XY^{X} by 𝒪 Y X\mathcal{O}_{Y^{X}}.

Exponentiability

Proposition

Let (X,𝒪 X)(X, \mathcal{O}_{X}) be a locally compact topological space, def. 1, and let (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be a topological space. The map ev:X×Y XYev : X \times Y^{X} \rightarrow Y given by (x,f)f(x)(x,f) \mapsto f(x) is continuous, where X×Y XX \times Y^{X} is equipped with the product topology 𝒪 X×Y X\mathcal{O}_{X \times Y^{X}} with respect to 𝒪 X\mathcal{O}_{X} and 𝒪 Y X\mathcal{O}_{Y^{X}} (notation 5).

Proof

By an elementary fact concerning continuous maps, it suffices to show that for any (x,f)X×Y X(x,f) \in X \times Y^{X}, and any U𝒪 YU \in \mathcal{O}_{Y} such that f(x)Uf(x) \in U, there is a U𝒪 X×Y XU' \in \mathcal{O}_{X \times Y^{X}} such that (x,f)U(x,f) \in U', and such that ev(U)Uev(U') \subset U.

To demonstrate this, we make the following observations.

1) Since ff is continuous, we have that f 1(U)𝒪 Xf^{-1}(U) \in \mathcal{O}_{X}.

2) Since (X,𝒪 X)(X, \mathcal{O}_{X}) is locally compact, we deduce from 1) that there is a V𝒪 X cV \in \mathcal{O}^{c}_{X} and a U𝒪 XU'' \in \mathcal{O}_{X} such that xUVf 1(U)x \in U'' \subset V \subset f^{-1}(U).

3) By 2) and by definition of 𝒪 X×Y X\mathcal{O}_{X \times Y^{X}} and 𝒪 Y X\mathcal{O}_{Y^{X}}, we have that U×M V,U𝒪 X×Y XU'' \times M_{V, U} \in \mathcal{O}_{X \times Y^{X}}, and that (x,f)U×M V,U(x,f) \in U'' \times M_{V, U}.

4) Let (x,f)U×M V,U(x',f') \in U'' \times M_{V, U}, Since xUx' \in U'', and since f(U)f(V)Uf'(U'') \subset f'(V) \subset U, the latter inclusion holding by definition of M V,UM_{V, U}, we have that f(x)Uf'(x) \in U. We deduce that ev(V×M V,U)Uev(V \times M_{ V, U}) \subset U.

By 3) and 4), we see that we can take UU' as the beginning of the proof to be U×M V,UU'' \times M_{V , U}.

Proposition

Let (X,𝒪 X)(X, \mathcal{O}_{X}), (Y,𝒪 Y)(Y, \mathcal{O}_{Y}), and (Z,𝒪 Z)(Z, \mathcal{O}_{Z}) be topological spaces. Let f:X×YZf : X \times Y \rightarrow Z be a continuous map, where X×YX \times Y is equipped with the product topology 𝒪 X×Y\mathcal{O}_{X \times Y} with respect to 𝒪 X\mathcal{O}_{X} and 𝒪 Y\mathcal{O}_{Y}. Then the map α f:XZ Y\alpha_{f} : X \rightarrow Z^{Y} given by xf xx \mapsto f_{x} is continuous, where f x:YZf_{x} : Y \rightarrow Z is given by yf(x,y)y \mapsto f(x,y), and where Z YZ^{Y} is equipped with the compact-open topology 𝒪 Z Y\mathcal{O}_{Z^{Y}}.

Proof

By an elementary fact concerning continuous maps, it suffices to show that for any xXx \in X, and any M A,U𝒪 Z YM_{A,U} \in \mathcal{O}_{Z^{Y}} such that f xM A,Uf_{x} \in M_{A,U}, there is a U𝒪 XU' \in \mathcal{O}_{X} such that xUx \in U', and such that α f(U)M A,U\alpha_{f}( U' ) \subset M_{A,U}.

To demonstrate this, we make the following observations.

1) Since f xM A,Uf_{x} \in M_{A,U}, we have, by definition of f xf_{x} and by definition of M A,UM_{A,U}, that f(x,a)Uf(x,a) \in U for all aAa \in A.

2) Since ff is continuous, we have that f 1(U)𝒪 X×Yf^{-1}(U) \in \mathcal{O}_{X \times Y}.

3) By 1), we have that {x}×Af 1(U)\{x\} \times A \subset f^{-1}(U).

4) Let 𝒪 X×A\mathcal{O}_{X \times A} denote the subspace topology on X×AX \times A with respect to 𝒪 X×Y\mathcal{O}_{X \times Y}. By 2), we have that f 1(U)(X×A)𝒪 X×Af^{-1}(U) \cap (X \times A) \in \mathcal{O}_{X \times A}.

5) By 3), we have that {x}×Af 1(U)(X×A)\{x \} \times A \subset f^{-1}(U) \cap (X \times A).

6) Since A𝒪 Y cA \in \mathcal{O}^{c}_{Y}, it follows from 4), 5), and the tube lemma that there is a U𝒪 XU' \in \mathcal{O}_{X} such that xUx \in U' and such that U×Af 1(U)(X×A)f 1(U)U' \times A \subset f^{-1}(U) \cap (X \times A) \subset f^{-1}(U).

7) We deduce from 6) that f(U×A)Uf(U' \times A) \subset U. This is the same as to say that f x(a)Uf_{x'}(a) \in U for all xUx' \in U'. Thus f xf_{x'} belongs to M A,UM_{A,U} for all xUx' \in U', which is the same as to say that α f(U)M A,U\alpha_{f}(U') \subset M_{A,U}.

We conclude that we can take the required UU' of the beginning of the proof to be the UU' of 6).

Proposition

Let (X,𝒪 X)(X, \mathcal{O}_{X}), (Y,𝒪 Y)(Y, \mathcal{O}_{Y}), and (Z,𝒪 Z)(Z, \mathcal{O}_{Z}) be topological spaces. Suppose that (X,𝒪 X)(X, \mathcal{O}_{X}) is locally compact. Let f:XZ Yf : X \rightarrow Z^{Y} be a continuous map, where Z YZ^{Y} is equipped with the compact-open topology 𝒪 Z Y\mathcal{O}_{Z^{Y}}. Then the map α f 1:X×YZ\alpha^{-1}_{f} : X \times Y \rightarrow Z given by (x,y)(f(x))(y)(x,y) \mapsto \big( f(x) \big)(y) is continuous, where X×YX \times Y is equipped with the product topology 𝒪 X×Y\mathcal{O}_{X \times Y}.

Proof

We make the following observations.

1) We have that α f 1=evτ(f×id)\alpha^{-1}_{f} = ev \circ \tau \circ (f \times id), where τ:Z Y×YY×Z Y\tau : Z^{Y} \times Y \rightarrow Y \times Z^{Y} is given by (f,y)(y,f)(f,y) \mapsto (y,f).

2) By Proposition 2, we have that evev is continuous.

3) It is an elementary fact that τ\tau is continuous.

4) Since ff is continuous, it follows by an elementary fact that f×idf \times id is continuous.

5) We deduce from 2) - 4) that evτ(f×id)ev \circ \tau \circ (f \times id) is continuous. By 1), we conclude that α f 1\alpha^{-1}_{f} is continuous.

Corollary

Let (X,𝒪 X)(X, \mathcal{O}_{X}) and (Y,𝒪 Y)(Y, \mathcal{O}_{Y}) be topological spaces. Suppose that (X,𝒪 X)(X, \mathcal{O}_{X}) is locally compact. Then (Y X,𝒪 Y X)(Y^{X}, \mathcal{O}_{Y^{X}}) together with the corresponding map evev defines an exponential object in the category Top of all topological spaces.

Proof

Follows immediately from Proposition 2, Proposition 3, and the fact that Y XY^{X} and evev are exhibited by the corresponding maps α \alpha_{-} and α 1\alpha^{-1}_{-} of Proposition 2 and Proposition 3 to define an exponential object in the category Set\mathsf{Set} of sets.

Remark

A proof can also be found in Aguilar-Gitler-Prieto 02, prop. 1.3.1, or just about any half-decent textbook on point-set topology!

However, the result is almost universally stated with an assumption that (X,𝒪 X)(X,\mathcal{O}_{X}) is Hausdorff which, as the proof we have given illustrates, is not needed.

Remark

We moreover have a homeomorphism Map(X,Map(Y,Z))Map(X×Y,Z)Map(X,Map(Y,Z))\cong Map(X\times Y,Z) if in addition XX is Hausdorff. See also convenient category of topological spaces.

Examples

Maps out of the point space

Example

(mapping space construction out of the point space is the identity)

The point space *\ast is clearly a locally compact topological space. Hence for every topological space (X,τ)(X,\tau) the mapping space Maps(*,(X,τ))Maps(\ast, (X,\tau)) exists. This is homeomorphic to the space (x,τ)(x,\tau) itself:

Maps(*,(X,τ))(X,τ). Maps(\ast, (X,\tau)) \simeq (X,\tau) \,.

Maps into metric spaces

If YY is a metric space then the compact-open topology on Map(X,Y)Map(X,Y) is the topology of uniform convergence on compact subsets in the sense that f nff_n \to f in Map(X,Y)Map(X,Y) with the compact-open topology iff for every compact subset KXK\subset X, f nff_n \to f uniformly on KK. If (in addition) the domain XX is compact then this is the topology of uniform convergence.

Loop spaces and path spaces

Example

(loop space and path space)

Let (X,τ)(X,\tau) be any topological space.

  1. The standard circle S 1S^1 is a compact Hausdorff space hence a locally compact topological space. Accordingly the mapping space

    XMaps(S 1,(X,τ)) \mathcal{L} X \coloneqq Maps( S^1, (X,\tau) )

    exists. This is called the free loop space of (X,τ)(X,\tau).

    If both S 1S^1 and XX are equipped with a choice of point (“basepoint”) s 0S 1s_0 \in S^1, x 0Xx_0 \in X, then the topological subspace

    ΩXX \Omega X \subset \mathcal{L}X

    on those functions which take the basepoint of S 1S^1 to that of XX, is called the loop space of XX, or sometimes based loop space, for emphasis.

  2. Similarly the closed interval is a compact Hausdorff space hence a locally compact topological space (def. \ref{LocallyCompactSpace}). Accordingly the mapping space

    Maps([0,1],(X,τ)) Maps( [0,1], (X,\tau) )

    exists. Again if XX is equipped with a choice of basepoint x 0Xx_0 \in X, then the topological subspace of those functions that take 0[0,1]0 \in [0,1] to that chosen basepoint is called the path space of (Xτ)(X\tau):

    PXMaps([0,1],(X,τ)). P X \subset Maps( [0,1], (X,\tau) ) \,.

Notice that we may encode these subspaces more abstractly in terms of universal properties:

The path space and the loop space are characterized, up to homeomorphisms, as being the limiting cones in the following pullback diagrams of topological spaces:

  1. loop space:

    ΩX Maps(S 1,(X,τ)) (pb) Maps(const s 0,id (X,τ)) * const x 0 XMaps(*,(X,τ)). \array{ \Omega X &\longrightarrow& Maps(S^1, (X,\tau)) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_{s_0}, id_{(X,\tau)})}} \\ \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau)) } \,.
  2. path space:

    PX Maps([0,1],(X,τ)) (pb) Maps(const x,id (X,τ)) * const x 0 XMaps(*,(X,τ)) \array{ P X &\longrightarrow& Maps([0,1], (X,\tau)) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Maps(const_x, id_{(X,\tau)})}} \\ \ast &\underset{const_{x_0}}{\longrightarrow}& X \simeq Maps(\ast,(X,\tau)) }

Here on the right we are using that the mapping space construction is a functor and we are using example 1 in the identification on the bottom right mapping space out of the point space.

References

  • Ralph H. Fox, On Topologies for Function Spaces , Bull. AMS 51 (1945) pp.429-432. (pdf)

  • Eva Lowen-Colebunders, Günther Richter, An Elementary Approach to Exponential Spaces, Applied Categorical Structures May 2001, Volume 9, Issue 3, pp 303-310 (publisher)

  • Marcelo Aguilar, Samuel Gitler, Carlos Prieto, sections 1.2, 1.3 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

  • Wikipedia entry

Revised on August 1, 2017 13:45:44 by Richard Williamson (84.202.247.109)