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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

7 votes
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Connected open sets in the topology generated by the collection of connected open sets

Take $\mathbb{Q}$ and add two points at infinity, $\infty_1$ and $\infty_2$, topologizing so that $\mathbb{Q} \cup \{\infty_i\}$ is the only open proper subset containing $\infty_i$ for $i \in \{1,2\} …
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4 votes
Accepted

Even covers and collectionwise normal spaces

As implied by Tyrone, the claim that a fully normal space is strongly collectionwise normal is true. See, for example, the result in H. J. Cohen's paper Sur un problème de M. Dieudonné (translated int …
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7 votes

On the continuity of a function given by evaluating compact subsets of smooth functions

I will answer the $\mathrm{C}^0$ case on a closed ball as in the other post. Let $f \colon \mathbb{R} \to \mathbb{R}$ be the continuous function $$ f(x) = \begin{cases} \frac{1}{2} &\text{if } x < \f …
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