Search Results

Another nice elementary example is the rational sequence topology. For every irrational number $x$ we pick a sequence $q(x)_n$ of rational numbers, all different, that converge to $x$ (in the usual to …

this paper by Howes gives a characterization of uniform spaces with a Lindelöf completion, but the characterization uses the derived uniformity, and notions like preparacompactness. I don't see as yet …

I like the proof from Alexander's subbase lemma. E.g. A proof here. That lemma also gives the compactness criterion in ordered spaces (completeness implies compactness).

As regards $\mathbb Q$ (your first remark), it is true that all countable metrisable spaces without isolated points are homeomorphic to $\mathbb Q$. If you want to omit metrisable, replace it by $\mat …

In Engelking's Theory of Dimensions, Finite and Infinite, Thm 3.3.10 (p. 200) proves the more general result If $f: X \to Y$ is a closed mapping from a normal space $X$ to a weakly paracompact normal …

To check that $f \to f \circ g$ is continuous in $f$ as a map $X^X \to X^X$ for a fixed $g \in X^X$: take a net $f_i \to f$ ($i \in I$, some directed set) in $X^X$ converging to $f \in X^X$. This mea …

this blog post refers to some papers with proofs. I've heard Robert Fokkink explain his proof (which is, quoting from this post) A linear map $\Bbb R^n \to \Bbb R^n$ can be understood to preserve or …

For me the most basic compact spaces are $\{x_n: n \in \Bbb N\} \cup \{x\}$ when $x_n \to x$, (the countable cofinite space is a special case), of course all finite spaces, and all ordered topological …

According to Engelking (exercise 3.4E, which is based on a paper by Arens): If $C(X,\Bbb R)$ (with the compact-open topology and $X$ Tychonoff) is first countable, then $X$ is hemicompact. A Hau …

It's an old (1981) theorem by Jan van Mill and Evert Wattel (see this paper) that a compact space has a continuous selection iff it is orderable. (So has a linear order whose order topology is the top …

This paper by Van Mill from 1981 gives a characterisation of $\Bbb Q \times \Bbb P$ (where $\Bbb P$ is a common notation for the irrationals) in Thm 5.3: If $X$ is separable metrisable and zero-di …

For a compact Hausdorff $X$ it is equivalent that $X$ is hereditarily Lindelöf or that $X$ is perfectly normal. Sketch: $X$ is hereditarily Lindelöf implies that every open set is an $F_\sigma$ (as $X …

$A=\{0\} \times [-1,1]^{\Bbb N}$ works as an example, e.g. For more info on $Z$-sets in $Q$, see Infinite-dimensional Topology by Jan van Mill.

Yes, there is. Let $I = \omega_1$ be the first uncountable ordinal, and let $P = \{0,1\}^I$ be the uncountable product of discrete spaces of 2 points. Let $S$, the so-called $\Sigma$-product be its s …

A continuous non-constant function from $[0,1]$ into $X$ with the cofinite topology exists iff $[0,1]$ has a partition into $\le |X|$ many disjoint closed non-empty subsets. This question discusses …