# Tagged Questions

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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### Given a basis for $\mathbb{R}$, show that it constructs the standard topology on $\mathbb{R}$

Let $q_1, q_2, ...,$ be the rational numbers enumerated. Consider the countable collection $$\mathcal{B} = \{ B_{\frac{1}{n}}(q_i) \ | \ i,n \in \mathbb{N} \}$$ of open balls centered at rational ...
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### Find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

I'd like to find a locally compact space $X$ with a subspace $A$ that is NOT locally compact. As from here, I know that if $A$ is closed and $X$ is Hausdorff, then $A$ is locally compact. Anyone ...
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### Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
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### Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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### what is the meaning of a power set of topological vector space?

Given a topological vector space, what is the power set of this space meaning? thanks a lot. and I really appreciate if a straightforward and simple explanation of topological vector space is ...
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### Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
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### What topological group is $\mathbb R/\mathbb Z$?

The integers $\mathbb Z$ are a normal subgroup of $(\mathbb R, +)$. The quotient $\mathbb R/\mathbb Z$ is a familiar topological group; what is it? I've found elsewhere on the internet that it is ...
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### Proof of commutativity for topological I-semigroups

A topological semigroup on a closed interval I and order topology is called I-semigroup if 1 acts as an identity and 0 as an annihilator. I've seen in several articles that such I-semigroups should ...
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### The fundamental group of a topological group is abelian [duplicate]

I want to show the fundamental group of a topological group is abelian. In fact, the question says the topological group is path connected. I do not know where I should use path-connectedness. I think,...
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### Neighbourhood base about a point $p$ of a topological group

I am reading topological groups from Van der Waerden. The conventions followed in this book are these. An open set that contains the point $p$ is called an open neighbourhood of $p$. Any set which ...
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### Non-solvable, closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$

It is mentioned here that non-solvable closed subgroups of $\mathrm{PSL}(2,\mathbb{R})$ are either the entire space or discrete. My question is this: Is there any easy proof of this, or do any of you ...
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### Integral of $|f|$ outside a compact set

Let $G$ be a locally compact group. Given $f\in L^1(G)$ and $\epsilon>0$, how to show that there is a compact set $K\subset G$ such that $\int_{G\setminus K}|f|<\epsilon$?
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### A semitopological qroup which is also quasitopological.

$(G,\mathcal T)$ is a semitopological group with $$(\forall x\in G)(\forall U\in \mathcal T)(\exists V \text{ neighborhood of }1)(x\notin U^cV)$$ Then $G$ is quasitopological (ie inverse is continuous)...
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### Homeomorphism between Space and Product

Do there exist examples of non-empty, infinite spaces X not equipped with the discrete topology for with $X \cong X \times X$?
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### Finite Haar Measure if and only if Compact

This is an exercise from a book: Let $G$ be a locally compact group with Haar measure $\mu$. $\mu(\{e\})>0$ if and only if $G$ is discrete. $\mu(G)<\infty$ if and only if $G$ is compact. I ...
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### Set of $x$ such that $h \mapsto hx$ is proper

Let $X$ be a locally compact second countable space, and $G$ a locally compact second countable group wich operates continuously on $X$. If $x \in X$, let $\rho_x : g \mapsto gx$. I would like to know ...
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### universal nonabelian divisible group

For this post, a group $G$ shall be referred to as generally divisible, in case $\forall{x\in G:}~\forall{n\in\mathbb{N}^{\times}:}~\exists{y\in G:}~y^{n}=x$. Note. Here is no commutativity ...
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### Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
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### Two basic questions about topological group theory

For a topological group, I'd like to know whether 1.there exist a topological group G which is a Hausdorff space but does not satisfies the first countable axiom or 2.there exist a topological ...
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### The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
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### Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
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### Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
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### Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
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### Homomorphism Theorem

Let $f$ be an open homomorphism from a topological group $G$ onto a topological group $H.$ We denote $K=Ker(f).$ How can I prove that $\bar f:G/K→H$ is a homeomorphism? I tried to prove it is ...
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### Can one visualise the dual groups to Cantor groups?

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the ...
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### looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
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### Exactness of completion of topological abelian groups

Let $0\to G^{\prime}\to G\to G^{\prime\prime}\to 0$ be an exact sequence of abelian groups. Suppose $G$ is a topological group and then give topologies on $G^{\prime}$ and $G^{\prime\prime}$ which are ...
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### Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it first-...
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### A question related to “Topology induced by the completion of a topological group”

I am sorry if I mistake the answer posted on the question Topology induced by the completion of a topological group. Stated as in that thread, let $G$ be a topological abelian group with a countable ...