When is a k-space locally compact?

Question

We're looking at the possible cardinal sequences of LCS (locally compact, Hausdorff, scattered) spaces, which has led us to think about taking a quotient of a locally compact, scattered space.

A k-space (compactly generated Hausdorff space) is a quotient space of a disjoint union of compact Hausdorff spaces. Every locally compact Hausdorff space is a k-space, but not every k-space is locally compact. However, every k-space is a quotient of a locally compact Hausdorff space.

1. Is there a general condition for a k-space to be locally compact?

2. What if we assume the space is a quotient of an LCS space?

3. Is there a condition that also guarantees a k-space is an LCS space?

Looking through Engelking and Google I couldn't find anything.

Answer 1

$\DeclareMathOperator\colim{colim}\DeclareMathOperator\CS{CS}\DeclareMathOperator\CSN{CSN}$The examples considered in algebraic topology are usually like $\mathbb{R}^\infty=\colim_n\mathbb{R}^n$ or $O(\infty)=\colim_n O(n)$ (the infinite orthogonal group) or infinite-dimensional CW complexes. These can usually be written as the colimit of locally compact Hausdorff spaces $X_n$ and closed inclusions $X_n\to X_{n+1}$, where $X_n$ typically has empty interior in $X_{n+1}$. In this context, a standard lemma says that every compact subset $K\subseteq X$ is contained in some $X_n$ and so has empty interior in $X_{n+1}$ (and therefore also has empty interior in $X$). Thus, examples of this type will never be locally compact.

One could consider the class $\mathcal{C}$ of k-spaces in which every compact set has empty interior, which is a kind of opposite to local compactness. I think that this class of spaces contains most of the popular examples and has good closure properties.

On the other hand, very many spaces can be written as a quotient of an LCS space. Indeed, let $\CS(X)$ be the set of continuous maps $\mathbb{N}\cup\{\infty\}\to X$ (i.e. pairs consisting of a convergent sequence and its limit). We can give $\CS(X)$ the discrete topology and the space $\CSN(X)=\CS(X)\times(\mathbb{N}\cup\{\infty\})$ the product topology. This makes $\CSN(X)$ into an LCS space, with an evident surjective evaluation map $\epsilon\colon \CSN(X)\to X$. The space $X$ is said to be sequential if $\epsilon$ is a quotient map. (The definition is usually phrased differently, but easily seen to be equivalent.) One can check that if $X$ is a k-space and every compact subspace is metrisable then $X$ is sequential. This covers most examples typically considered in algebraic topology.

My guess is that the space $\beta(\mathbb{N})$ (the Stone–Čech compactification of the naturals) is not a quotient of an LCS space, but I have not checked that.

Answer 2

For first-countable regular spaces (which form an important subclass of LCS-spaces) there exists a simple characterization: a first-countable regular space $X$ is not locally countably compact if and only if $X$ contains a closed subspace, homeomorphic to the subspace $$\mathbb L=\{(0,0)\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\}$$ of the Euclidean plane.

This characterization implies that a first-countable paracompact space $X$ is locally compact if and only if $X$ contains no closed subspaces, homeomorphic to $\mathbb L$.