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Don't pick a convergent subsequence: pick a point in $ \cap_{n \in \mathbb{N}} \overline{\{ x_n,x_{n+1}, \dots \}}$ (the overline meaning closure) which is non empty as an intersection of decreasing f …

Note : In order to define the alexandrov compactification of $X$ you have to take the algebra of function converging at infinity (not just those converging to $0$) If by a compactification of $X$ you …

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 …

I'm assuming that by "compact" you only mean which satisfies the finite cover properties, and not compact and Hausdorff. in this case the following produces a counterexample: Let $X_1$ be $\mathbb{R} …

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 …

The answer is yes. It appears for example as lemma C3.1.15 in Johnstone's sketches of an elephant. Roughly, it can be proved by working in the internal logic of the target (hence assuming that the t …

I have thought about this recently, and here is I think the best constructively valid statement one can extract from Brouwer fixe point theorem (framework : internal logic of an elementary topos, real …

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 …

For short, the exponential $(X,Y)$, characterized by the usual universal properties: morphisms from any locale $Z$ to $(X,Y)$ are functions from $X \times Z$ to $Y$, exists for all $Y$ if and only if …

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 …

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 …

I confirme my comment : $X$ is the stone-cech compactification of a discrete space if and only if $X$ is compact, haussdorf, extremally disconected, and has a dense set of open points. here is a ske …

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 …

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 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 …