# Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

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### Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### Finite-Dimensional Homogeneous Contractible Spaces

Suppose that $X \subset \mathbb{R}^n$ is compact, homogeneous and contractible (and thus connected). Does $X$ have to be a point? I couldn't think of a non-trivial example, and there isn't a ...
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### Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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### What is the structure preserved by strong equivalence of metrics?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are (topologically) equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ ...
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### Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?

Let $X$ be a nontrivial topological space, $I$ be a infinite set, we can endow $X^I$ (the set of all functions $I\to X$) with either the product topology or the box topology. We know that the box ...
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### Is there a topological space of cardinality $\kappa$ containing all metric spaces of cardinality $\kappa$?

It is known that all the countable metrizable spaces embed in $\mathbb{Q}$. Then a natural question is: are there an uncountable cardinal $\kappa$ and a topological space $X$ of cardinality $\kappa$ ...
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### Exercise in Engelking's book regarding a disconnected space.

This is related to 6.3.24 in Engelking's Topology book. The Hilbert space $H$ is the set of sequences $(x_i) \in \mathbb R ^\omega$ such that $\|x\|=\sum _{i=1} ^\infty x_i ^2<\infty$, with ...
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### Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset\Bbb{R}$ intersect every uncountable $\mathcal{F}_{\sigma}$ set. $B\subset \Bbb{R}$ is said to have the property of Baire if $B=U\triangle M$ where $U$ is open and $M$ is meager. Does ...
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### Is there a path-connected, "anti-convex" subset of $\mathbb R^2$ containing $(\mathbb R\smallsetminus \mathbb Q)^2$?

This question was firstly asked in Mathematics Stack Exchange. Getting no answer, I copied it to Math Overflow, as Moishe Kohan commented. For a vector space $V$ over $\mathbb R$, I say a subset $S$ ...
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### A natural topology on a field

I can endow any field with a natural topology in the following way. Given a polynomial $f\in K[X]$, I denote by $\mathcal{O}(f)=\{x\in K\mid\exists y\in K^{\times}\ f(x)=y^2\}$, i.e. the set of ...
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