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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 …

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.

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 …

You're probably right about the statement of the theorem. Indeed, many textbooks (especially older ones) call nets "generali(s/z)ed sequences" I think it should say: Given a net $x: \mathcal{I} \ …

If $K$ is compact in $Z$ and $U$ open in $Y$, then $K$ is still compact as a subset of $X$ as well (compactness is absolute, or maybe use that $i[K]$ is compact where $i: Z \to X$ is the continuous c …

Your corrected version of exercise 2: Suppose $X$ is $T_4$ and $X=\bigcup \mathcal{U}$ where all $U \in \mathcal{U}$ are open and metrisable, and $\mathcal{U}$ is locally finite. Then $X$ is metr …

The obvious (to me) counterexamples are $\beta \mathbb{N}$ and $\beta \mathbb{R}$ ( the Čech-Stone compactifications) which are non-completely normal compactifications (classic fact, see Engelking or …

For boundedness of sets the statement is false. The Wikipedia quote is for linear operators. A counterexample for sets: $X=\mathbb{R}^\omega$ in the product topology is a metric locally convex TVS. …

A simple solution: if $X$ is second countable, let $D=\{d_n : n =1,2,3,\ldots\}$ be a dense subset of $X$ and define $$\mu(A)= \sum_{n:d_n \in A}\frac{1}{2^n}$$ for all subsets of $X$. Then clearly …