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EDIT: I apologize for the confusion by which I originally framed this question for normal spaces, where it has an uninteresting answer (thanks to those who pointed this out). Hope I've got it right n …

In a Hausdorff but not regular space, collapsing certain closed sets to a point may produce a non-Hausdorff space. Does there exist a term for closed sets one may collapse and still have a Hausdorff …

The following topological property arose in the context of my friend Don Hadwin's operator theory research, and he asked me to ask here if the property occurs in the literature and has a name. Proper …

I believe you can grow your example B into a counterexample. Stage 0 is $\emptyset$ Stage 1 is $\{p\}$. Stage 2 is your $B$: you've added a copy of ${\Bbb N}$ for each point newly added in the pre …

The counterexamples so far depend on AC, but one can have such spaces just from ZF. In particular, instead of an ultrafilter on ${\Bbb N}$ as in your example $B$, one can use the filter of subsets wi …

As for $Q_2$, take $X \times S^1$ with $X$ discrete. The one-point Lindeloffication follows along the same lines as above. Of course any one-point compactification is a fortiori a one-point Lindelof …

The impatient reader can skip my attempt at motivation and go straight my "Question formulations for the impatient." In a failed(?) attempt at discovering something new, some years ago I toyed with …

Given a(n infinite) set $S\subset {\Bbb Z}[x]$ (integer polynomials), write $R_S$ for the topological closure of the set of all complex roots of all $p\in S$. Then write $\hat{S}$ for the set of all …

Given a non-Hausdorff space $X$, one can form a graph $G_X$: vertices the points of $X$, edges indicating point pairs not separated by open sets. Up to graph-theoretically (but not topologically) iso …

Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \be …

I am given to understand that the celebrated open problem (MLC) of the Mandelbrot set's local connectness has broader and deeper significance deeper than some mere curiosity of point-set topology. …

Start with the Hilbert cube $H=I^\omega$, thinking of its coordinates as written in ternary expansion. Construct subsets $S_n$ by removing points from $H$ if for any $m$, at least $n$ of the coordi …

Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space topolo …

Is every sigma-algebra the Borel algebra of a topology? inspires the present question which asks for less. Question: Given a $\sigma$-algebra $\mathcal A$ on a set $X$, does there exist a topology $\m …

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of metric spaces is metrizable, simply by rescaling or chopping off the individual metrics to have diameter at most one, and …