# Questions tagged [gn.general-topology]

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

582 questions
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### Is an Abelian topological group compact if it is complete and Bohr-compact?

A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff. A topological group $G$ is Bohr-compact if it admits ...
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### A strict directed colimit of Hausdorff locally-convex spaces that is not Hausdorff

We work in the category of locally-convex spaces (morphisms are the continuous linear maps). Let $\Lambda$ be a directed set, for every $\lambda \in \Lambda$ let $V_{\lambda}$ be a locally-convex ...
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### What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
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### Bundle structures on spheres

Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover,...
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### Basic calculus on topological fields

Let $K$ be a a topological field (I am mainly interested in the cases when K is either an ordered field or a valued field, e.g. $K = \mathbb Q$ or $\mathbb Q_p$). 1) Let $f: K^n \to K$ be a ...
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### Counting loops in degree: 1 or 2?

Here's what seems to be an annoying technicality when dealing with loops in graphs. In the literature on expander graphs (and surely not only), it seems to be the convention that a loop at vertex $v$ ...
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### Homeomorphism between evenly spaced integer topology and the rationals

The evenly spaced integer topology is countable, metrizable, and has no isolated points, and hence is homeomorphic to the rationals with the order topology. But what is an explicit construction for ...
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### Topology on the space of Borel measures

Let $B$ be the set of all measures $\phi$ of $\mathbf{R}^{n}$ such that every open set is $\phi$-measurable (sometimes these measures are called Borel measures). Note the measures in $B$ are ...
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### Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...
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### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
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### algebraic structure of Integral Steenrod squares

It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations $$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$ In the case where $a$ is odd, one can define an ...
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### Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem. Let $G$ be a locally compact amenable ...