# Questions tagged [gn.general-topology]

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

2,918 questions
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### Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem?

According to Brouwer's fixed point theorem, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point. However, these fixed points cannot be chosen ...
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### When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE. Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...
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### Can the real line be embedded in a space $X$ such that all the nonempty open subsets of $X$ are homeomorphic?

The question is in the title: Q1: Is there a topological space $X$ containing a copy of the real line and having the property that all the nonempty open subsets of $X$ are homeomorphic? Let us ...
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### Identification of ultrafilters with measures

We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$. Now my question is which ...
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### Path Metric Topology

Is there an example of a metric space $(X,d)$ whose corresponding path metric, $d^\prime$ generates a strictly finer topology compared to the topology generated by $d$?
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### Maxed-out Hausdorff metric

Let $(Y,d)$ be a non-degenerate compact metric space, and let $d_H$ be the Hausdorff metric (https://en.wikipedia.org/wiki/Hausdorff_distance) on $K(Y)$ generated by $d$. Here $K(Y)$ is the set of ...
1answer
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### Connected spaces where every dense set is large

Let $\kappa >\aleph_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every dense set $D\subseteq X$ we have $|D|=|X|$?
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### Partitions of unity in constructive mathematics

Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics? In particular, under what circumstances can we construct a partition of unity subordinate to ...
3answers
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### Metrizable subspaces of separable spaces

Are metrizable subspaces of separable spaces separable? Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.
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### Compactness of the Fell topology and local compactness

Given some topological space $\mathbf{X}$, we consider the Fell topology on the set of closed subsets of $\mathbf{X}$. This is generated by sets of the form $I_U = \{A \mid A \cap U \neq \emptyset\}$ ...
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