Todd Trimble
core-compact spaces are exponentiable

Contents

Introduction

Recall that a space XX is locally compact if each point has a neighborhood basis consisting of compact neighborhoods.

Somewhat less known but slightly more general is the class of spaces called “core-compact spaces”:

Definition

A topological space is core-compact if, for every open neighborhood VV of a point xx, there is a smaller open neighborhood UU of xx, such that every open covering of VV admits a finite subcover of UU. We write UVU \ll V (pronounced “UU is well below VV”) to indicate that this covering condition holds.

Core-compactness is equivalent to the statement that every open VV is the union of open sets UU such that UVU \ll V.

The goal of this article is to give a succinct proof that core-compact spaces are exponentiable in TopTop, so that in particular locally compact spaces (with no further regularity conditions) are exponentiable in TopTop. This proof is extracted from a nice article by Escardó and Heckmann, who show moreover that exponentiable spaces in TopTop are precisely the core-compact spaces (which is a matter we won’t go into here).

Direct description of the exponential topology

For given topological spaces XX and YY, we let Top(X,Y)Top(X, Y) be the set of continuous functions f:XYf: X \to Y. The theory developed by Escardó and Heckmann shows that there is a “best” candidate for an exponential topology on Top(X,Y)Top(X, Y), which they call the Isbell topology but which is also known as the “natural topology” or the “topology of continuous convergence”. Whatever it is called, it is uniquely characterized by the following property:

In the case where YY is the Sierpinski space 2={0,1}\mathbf{2} = \{0, 1\} (where 11 is open but 00 is not), Top(X,2)Top(X, \mathbf{2}) is in natural bijection with the set 𝒪(X)\mathcal{O}(X) of open sets of XX, i.e., the topology of XX. Under this isomorphism the natural topology on Top(X,2)Top(X, \mathbf{2}) corresponds to a topology also known as the Scott topology on 𝒪(X)\mathcal{O}(X):

Definition

A subset 𝒰𝒪(X)\mathcal{U} \subseteq \mathcal{O}(X) is called Scott-open if

  • it is upward-closed: if UVU \subseteq V are open sets of XX and U𝒰U \in \mathcal{U}, then V𝒰V \in \mathcal{U};

  • for every V𝒰V \in \mathcal{U} there exists U𝒰U \in \mathcal{U} such that UVU \ll V.

Then for arbitrary spaces YY, the natural topology on Top(X,Y)Top(X, Y) may be explicitly described as the topology generated by sets of the form

F(𝒰,V){XfY:f 1(V)𝒰}F(\mathcal{U}, V) \coloneqq \{X \stackrel{f}{\to} Y: f^{-1}(V) \in \mathcal{U}\}

where VV ranges over open sets of YY and 𝒰\mathcal{U} over Scott-open subsets 𝒰\mathcal{U} of 𝒪(X)\mathcal{O}(X).

We let Y XY^X denote the space whose underlying set is Top(X,Y)Top(X, Y) and whose topology is the natural topology thus described. The theorem we prove may be stated as follows:

Theorem

If XX is core-compact and YY is any space, then the evaluation map Y X×XYY^X \times X \to Y is continuous, and realizes (via the Yoneda lemma) a natural isomorphism Top(,Y X)Top(×X,Y):Top opSetTop(-, Y^X) \cong Top(- \times X, Y): Top^{op} \to Set. Thus, Y XY^X satisfies the universal property of an exponential in TopTop.

As we said, the proof we give is a very whittled-down rendition extracted from the much more thorough account given by Escardó and Heckmann. It is based on two lemmas:

Lemma

For any spaces A,YA, Y, and XX (without assuming XX is locally compact), if g:A×XYg: A \times X \to Y is continuous, then its transpose g :AY Xg^\wedge: A \to Y^X is continuous.

Lemma

For any space YY, the evaluation map ev:Y X×XYev: Y^X \times X \to Y is continuous if XX is core-compact.

It follows that if h:AY Xh: A \to Y^X is continuous, then so is the inverse transpose

h =(A×Xh×1Y X×XevY).h^\vee = \left(A \times X \stackrel{h \times 1}{\to} Y^X \times X \stackrel{ev}{\to} Y\right).

As the operations gg g \mapsto g^\wedge and hh h \mapsto h^\vee are mutually inverse, this establishes the natural isomorphism Top(A,Y X)Top(A×X,Y)Top(A, Y^X) \cong Top(A \times X, Y) of the theorem.

Proofs of the lemmas

First we establish Lemma 1.

Proof

Suppose g:A×XYg: A \times X \to Y is continuous, and let F(𝒰,V)F(\mathcal{U}, V) be a typical generating element of the topology of Y XY^X. We are to show that

(g ) 1(F(𝒰,V)) = {a:g (a)F(𝒰,V)} = {a:g (a) 1(V)𝒰}\array{ (g^\wedge)^{-1}(F(\mathcal{U}, V)) & = & \{a: g^\wedge(a) \in F(\mathcal{U}, V)\} \\ & = & \{a: g^\wedge(a)^{-1}(V) \in \mathcal{U}\} }

is open in AA. Fix an element a(g ) 1(F(𝒰,V))a \in (g^\wedge)^{-1}(F(\mathcal{U}, V)); then

g (a) 1(V)={x:(a,x)g 1(V)}.g^\wedge(a)^{-1}(V) = \{x: (a, x) \in g^{-1}(V)\}.

Consider the collection of all pairs (W,U)(W', U') where UU' is open in XX and WW' is an open neighborhood of aa such that W×Ug 1(V)W' \times U' \subseteq g^{-1}(V). The collection of such UU' covers g (a) 1(V)g^\wedge(a)^{-1}(V). Since g (a) 1(V)𝒰g^\wedge(a)^{-1}(V) \in \mathcal{U} and 𝒰\mathcal{U} is Scott-open, there are finitely many such UU', say U 1,,U nU_1, \ldots, U_n, that cover some U𝒰U \in \mathcal{U}, and we have corresponding open neighborhoods W 1,,W nW_1, \ldots, W_n of aa such that W i×U ig 1(V)W_i \times U_i \subseteq g^{-1}(V) for i=1,,ni = 1, \ldots, n.

We claim the neighborhood W=W 1W nW = W_1 \cap \ldots \cap W_n of the point aa is included in (g ) 1(F(𝒰,V))(g^\wedge)^{-1}(F(\mathcal{U}, V)). Indeed, suppose bWb \in W; we must show g (b) 1(V)𝒰g^\wedge(b)^{-1}(V) \in \mathcal{U}. But since 𝒰\mathcal{U} is Scott-open, it is upward closed, and so it suffices to verify that

Ug (b) 1(V)={x:(b,x)g 1(V)}.U \subseteq g^\wedge(b)^{-1}(V) = \{x: (b, x) \in g^{-1}(V)\}.

But if xUx \in U, then xU ix \in U_i for some ii and of course bW ib \in W_i, so (b,x)W i×U ig 1(V)(b, x) \in W_i \times U_i \subseteq g^{-1}(V) and the claim is proven.

Now we prove Lemma 2.

Proof

Let fY X,xXf \in Y^X, x \in X; let VV be an open neighborhood of f(x)f(x). Then f 1(V)f^{-1}(V) is an open neighborhood of xx, and since XX is core-compact, there is an open neighborhood UU of xx such that Uf 1(V)U \ll f^{-1}(V). Put

𝒰={WopeninX:UW}\mathcal{U} = \{W\; open\; in \; X: U \ll W\}

Then it is enough to show

  1. 𝒰\mathcal{U} is Scott-open (and hence F(𝒰,V)F(\mathcal{U}, V) is open in Y XY^X);

  2. (f,x)F(𝒰,V)×U(f, x) \in F(\mathcal{U}, V) \times U;

  3. F(𝒰,V)×Uev 1(V)F(\mathcal{U}, V) \times U \subseteq ev^{-1}(V).

The only tricky part is 1. For 2. it is obvious that fF(𝒰,V)f \in F(\mathcal{U}, V) since f 1(V)𝒰f^{-1}(V) \in \mathcal{U}, i.e., Uf 1(V)U \ll f^{-1}(V). For 3., if gF(𝒰,V)g \in F(\mathcal{U}, V) and xUx \in U, then Ug 1(V)U \ll g^{-1}(V), whence xUg 1(V)x \in U \subseteq g^{-1}(V) and then ev(g,x)=g(x)Vev(g, x) = g(x) \in V, as required.

For 1., it is obvious that if UWU \ll W and WWW \subseteq W', then UWU \ll W', so 𝒰\mathcal{U} is upward-closed. Now we need to check that for all WW' in 𝒰\mathcal{U} there exists WW in 𝒰\mathcal{U} such that WWW \ll W'. By core-compactness,

W= WWW= WW VWV= W:VWWVW' = \bigcup_{W \ll W'} W = \bigcup_{W \ll W'} \bigcup_{V \ll W} V = \bigcup_{\exists W: V \ll W \ll W'} V

The covering of WW' by open sets VV such that W:VWW\exists W: V \ll W \ll W' is a collection that is closed under finite unions, and so since UWU \ll W', we have that UVU \subseteq V for some such VV. If UVWU \subseteq V \ll W, then UWU \ll W, so indeed there is W𝒰W \in \mathcal{U} such that WWW \ll W'. This completes the proof that 𝒰\mathcal{U} is Scott-open and the proof of the lemma.

Remark

It is not hard to deduce that if XX is core-compact, then the collections of the form

𝒰 U={W:UW}\mathcal{U}_U = \{W: U \ll W\}

form a base of the Scott topology on 𝒪(X)\mathcal{O}(X). In that case, the exponential = natural topology on Y XY^X may be described as generated by sets of the form

F U,V{XfY:Uf 1(V)}F_{U, V} \coloneqq \{X \stackrel{f}{\to} Y: U \ll f^{-1}(V)\}

where U,VU, V range over open sets of X,YX, Y, respectively. This bears a closer resemblance to the classical compact-open topology than our first abstract description of the natural topology on Top(X,Y)Top(X, Y) in terms of the Scott topology on 𝒪(X)\mathcal{O}(X).

Reference

Created on July 31, 2017 at 11:57:28 by Todd Trimble