# Questions tagged [gn.general-topology]

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

2,977 questions
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### If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?

If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we necessarily have $|X| =|\tau|$?
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### A special topological property

Let us say a topological space $X$ is a countable union of second countable spaces if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that the relative topology on $X_n$'s ...
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### Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?

Let $\mathbb{Q}$ be the rationals, and let $\tau$ be the Euclidean topology on $\mathbb{Q}$. Is there a topology $\tau' \subseteq \tau$ such that $(\mathbb{Q},\tau')$ is connected and $T_2$?
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### Approximation of the identity by simple functions

Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$. Can we concluded $X$ may be written by a ...
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### Are second-countable subsets of topological vector spaces metrizable?

Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable. Q. Does there ...
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### Approximation on separable topological space with size $\mathfrak{c}$

Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function. Q. Is it possible to find a sequence of ...
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### Connected $T_2$-space such that the open sets are closed under countable intersection

This question has a trivial starting point: If the open sets of $\mathbb{R}$ were closed under countable intersection, the Euclidean topology would be discrete because for all $x\in\mathbb{R}$ we have ...
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### A point concerning Fremlin's example on Borel sets in non-separable Banach spaces

Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$. $~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology. $~~\mathcal{M}$= The sigma algebra ...
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### Comparing two $\sigma$-algebras

Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$. Q. For which ...
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### $T_2$-spaces where all non-empty open sets are homeomorphic

We say that a $T_2$-space $(X,\tau)$ has homeomorphic open sets if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$. The rationals with the ...
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### A compact set in the plane with small sum-set and large projections

Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane? Remark. The ...
Suppose that $X$ is a metric space and let $C_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C_k(X)$ is a topological vector space. Let ...