Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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2
votes
1answer
92 views
+100

Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
1
vote
0answers
63 views
+100

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
3
votes
0answers
37 views
+500

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
0
votes
0answers
6 views
+300

Distributivity of group topologies on a Prüfer group

Let $p$ be a prime number and $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that $(\Bbb Z_{p^\infty},\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ ...