5

We start with some preliminary definitions (necessary because there is not much literature on this):

  • a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of $\mathbb{R}^n$ for some $n$.
  • Given a topological space $X$, we say that a subset $U$ of $X$ is numerically open if $\varphi^{-1}(U)$ is open for every test map $\varphi$.
  • A topological space $X$ is said to be numerically generated if a subset $U$ of $X$ is open if and only if it is numerically open.
  • Given any topological space $X$, we define the numericalization of $X$, denoted $X^\#$, to be the set underlying $X$ equipped with the final topology induced by the family of test maps into $X$.
  • The numerically generated product of two numerically generated spaces $X$ and $Y$ is the numericalization of the product space $X\times Y$ (henceforth simply denoted $X\times Y$).
  • The numerically generated mapping space of the numerically generated spaces $X$ and $Y$, denoted $Map(X,Y)$, is the collection of all continuous maps from $X$ to $Y$ equipped with the numericalization of the compact-open topology.
  • The category $\mathbf{Num}$ is the full subcategory of $\mathbf{Top}$ with objects as numerically generated spaces; this has as coproduct the disjoint union with the disjoint union topology (it is not difficult to show that the disjoint union of numerically generated spaces is numerically generated) and product as the numerically generated product.

I wish to show that $\mathbf{Num}$ is a cartesian-closed category. Ultimately, this simply boils down to showing that the numerically generated mapping space is an exponential object in $\mathbf{Num}$.

Naturally, I would want to use the typical bijection $h: Hom(X\times Y,Z)\rightarrow Hom(X,Z^Y)$ given by currying, but it remains to show that given a continuous map $f:X\times Y\rightarrow Z$, that the map $g:X\rightarrow Map(Y,Z)$ defined by $g(x)=f(x,-)$ is continuous (as well as that $f(x,-)$ is continuous too). However, I'm tripping up on showing this.

Martin Sleziak
  • 53,687
Hayden
  • 16,737
  • 1
    Have you tried the emulate the usual arguments that go into showing that this is true for compactly generated spaces? – Piotr Pstrągowski Mar 14 '14 at 18:11
  • @PiotrPstragowski I have not, since I don't know the usual arguments; this is my first introduction to algebraic topology and we're not using compactly generated spaces, but I'll try to take a look at the proof of that subcategory. – Hayden Mar 14 '14 at 18:17
  • I've never seen this definition before! Where did you encounter it? – Zhen Lin Mar 14 '14 at 19:57
  • Professor Barwick at MIT; we're studying the Fundamental Groupoid & the Postnikov Tower in our Topology Seminar. Having done some searching before, slight mention of the topic is seen in some higher level algebraic topology papers, but nothing more than that. – Hayden Mar 14 '14 at 20:00
  • 1
    There is quite a lot of literature in this area. See for example Booth, P.I., Tillotson, J., "Monoidal closed categories and convenient categories of topological spaces". Pacific J. Math. 88 (1980) 33-53. See also Section 5.9 of my book "Topology and Groupoids". It is more usual to take the test maps from some compact Hausdorff spaces, so it is interesting to see this definition. – Ronnie Brown Mar 15 '14 at 12:47
  • @RonnieBrown It ends up that a space $X$ is numerically generated if and only if it has the final topology induced from the family of continuous maps from $I=[0,1]$ to $X$, so the above definition of numerically generated spaces is in fact the same as using test maps from a compact Hausdorff space (in this case the unit interval). – Hayden Mar 20 '14 at 18:21
  • I'm coming back to your last comment, where you are indicating that a subset is numerically open iff it has a closed preimage under every map from $I$ to $X$. I can show that if it's numerically open, then its preimage in $I$ must be open, since we can factor $t:I\to X$ through some open interval. But how to prove the converse? – Stefan Hamcke May 21 '15 at 17:59

1 Answers1

3

I know this answer is coming with a few years of delay, but there is a paper by Escardó, Lawson and Simpson called comparing Cartesian closed categories of (core)compactly generated spaces that gives a very general result. I'll highlight the main parts here for your convenience.

Let $\mathbf{Top}$ be the category of topological spaces and let $I$ be a small full subcategory of $\mathbf{Top}$.

Definition: A topological space $X$ is said to be $I$-generated, if it is a colimit of objects in $I$. Equivalently, $X$ is $I$-generated if it has the final topology with respect to the family of all maps $f\colon C\to X$ from objects in $I$ to $X$.

Proposition: Assume that all the objects in $I$ are core-compact (aka exponentiable in $\mathbf{Top}$). Moreover, assume that finite products of objects in $I$ are $I$-generated. Then, the category of $I$-generated spaces is Cartesian closed.

In particular, numerically generated spaces are Cartesian closed. The way I like to see this is that numerically generated spaces coincide with $\Delta$-generated spaces, where we take $\Delta$ to be the full subcategory of $\mathbf{Top}$ spanned by the geometric realisations of the standard simplices. Then, since realisation commutes with products of standard simplices, the claim follows.

Stefano Nicotra
  • 371