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Here is I think a counter-example to the precise proposed formula in the question. Take $X= \mathbb{Q}$ with the discrete topology, with $Y = \mathbb{R}$ withe its usual topoly and the map $f:X \to Y$ …

When $X$ is compact, it is discrete (hence not compact unless it is finite). For any function $f:X \to \{0,1\}$, then both $K_0 = f^{-1}(0)$ and $K_1=f^{-1}(1)$ are compact subset of $X$, and so the s …

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases: One first generalize …

If $X$ is locally compactly generated then $X$ is compactly generated because every locally compact space is compactly generated. So given $f:X \to Y$ a map such that $f\circ i$ is continuous for ever …

I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts: The category of compact Hausdorf topolog …

Let me first clarify some confusion in the comments to the original question. To be clear : I'm not at all saying the persons making them were confused, as far as I can tell all the comments were corr …

So the short answer is that there is no non-empty projective locales for essentially any reasonable class of epimorphisms you can think of (except maybe proper maps). The problem is that there exists …

I like to call this result the localic Baire category theorem, and it plays essentially the same role as Baire category theorem: it lets you "construct" object by showing that some spaces are non-empt …

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for …

I claim that the following result have constructive* proof: 1) Let $f : [0,1] \rightarrow \mathbb{R}$ be a uniformly continuous function such that $f(0)\leqslant 0$ and $f(1) \geqslant 0$ then (as a …

Let me expand a bit my comment as this is a rather subtle property. As I said any locally compact Hausdorff topological space is a strongly hausdroff locally compact locales. (and under the axiom of …

I'm looking for a reference for the following result: Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over …

Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of t …

The answer is No. Rougly, because it is not a good idea to look at continuous $\mathbb{C}$ valued function on a space which is not completely Haussdorff as completely haussdorf is exactly the hypothes …

By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ca …