# 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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### Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
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### Polish topological group

A friend asked me to help him prove that the topological group $\mathrm{Homeo}(0,1)$ (homeomorphism of $(0,1)$ with the compact open topology) is Polish (that is, separable and completely metrizable). ...
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### Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
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### Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
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### Probability space associated with a compact group

Is the probability space associated with a compact group with Haar probability always a standard probability space? I recall seeing somewhere the fact that if the topology generating the Borel sigma ...
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### $G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint below....
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Each $q$ in the symplectic group $Sp(1)$, defines an operator on the imaginary quaternions: $\rho(q):Im\mathbb{H} \rightarrow Im\mathbb{H}$ by $\rho(q)(x)=qx\bar{q}$ We can see here that $\rho:Sp(1) ... 1answer 82 views ### What is the cokernel of$\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$? These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} \... 1answer 45 views ### Different definitions of topological group [duplicate] Recently I discovered the definition of topological group. So, topological group is an abstract group G endowed with topological structure such that the maps mult: G\times G\longrightarrow G and ... 1answer 29 views ### Topology on the group of autohomeomorphisms I am wondering whether the group of all autohomeomorphisms of a compact metric space can be given a reasonable topological group structure? (Preferably, can it be turned into a locally compact group?) ... 1answer 47 views ### Is there any standard terminology for the quotient of a topological group by the connected component of the identity? If G is any topological group, then the connected component of its identity is a closed normal subgroup H. It follows that G/H is a totally disconnected topological group. Often, G will be ... 1answer 237 views ### Group Structure on \Bbb R (\Bbb R,+) is a topological group. Is there any other group structure on \Bbb R such that it is still a topological group and this group is not isomorphic to (\Bbb R,+) ? Refer to Non-... 2answers 239 views ### Non-isomorphic Group Structures on a Topological Group Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ... 1answer 61 views ### Constructing Topological Groups [closed] In general, is there a way to construct topological groups? That is, given two topological groups X and Y, can I construct a topological group Z using X and Y in said construction? I have ... 1answer 75 views ### Unique Square Root Neighbourhood in Topological Group For a Lie Group \mathfrak{G} and any neighbourhood \mathcal{V}\subset\mathfrak{G} of the identity \mathrm{id}\in\mathfrak{G}, \exists neighbourhood \mathcal{U}\subset\mathcal{V} of \mathrm{... 1answer 208 views ### 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 ... 3answers 60 views ### 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 ... 1answer 80 views ### 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 ... 0answers 112 views ### 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 ... 1answer 47 views ### 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 ... 1answer 235 views ### 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 ... 2answers 389 views ### 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 ... 0answers 33 views ### 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 ... 1answer 2k views ### 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,... 1answer 154 views ### 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 ... 0answers 36 views ### 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 ... 1answer 36 views ### 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? 1answer 33 views ### 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)... 2answers 263 views ### 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? 1answer 364 views ### 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 ... 0answers 17 views ### 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 ... 0answers 30 views ### 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 ... 0answers 47 views ### 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 ... 1answer 70 views ### 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 ... 1answer 115 views ### 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 ... 0answers 44 views ### 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 ... 1answer 155 views ### 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 ... 1answer 102 views ### 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. ... 1answer 91 views ### 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 ... 1answer 103 views ### 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 ... 0answers 71 views ### 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 ... 1answer 82 views ### 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 ... 1answer 735 views ### 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-... 1answer 111 views ### 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 ... 1answer 52 views ### A question on Cauchy sequence in topological abelian group Let G be a topological abelian group. Recall that a Cauchy sequence (x_n) in G is defined to be a sequence such that for any neighborhood U of 0, there exists an integer N with x_n-x_m\... 0answers 83 views ### Abstract Fourier Analysis I am trying to prove that given a locally compact abelian (Hausdorf) topological group, the characters on it parameterize the multiplicative linear functionals on the banach * algebra L^1(G, d\mu) ... 2answers 273 views ### Topological groups question from Munkres This if from the 'Supplementary Exercises' at the end of Chapter 2 in Munkres' Topology. If A and B are subsets of (a topological group) G, let A \cdot B denote the set of all points a \cdot ... 2answers 97 views ### If f is a positive function and \int_{E}f d\lambda = 0 then \lambda (E) = 0 If f is a positive function and$$ \int_{E}f d\lambda = 0,$$then show that$\lambda (E) = 0$where$\lambda $is a Haar (Radon) measure. I know that if$f$is a positive function and$\int_{E}...
Let $E$ be a Hausdorff topological space, $G$ a homeomorphism group that acts on $E$ properly discontinous, i.e. $\forall e\in E$ exists a neighborhood $U$ of $e$ such that $gU\cap U = \emptyset$ for ...
Let $X$ be a topological group and let $D$ is a countable discrete closed subset of $X$. We also let $\mathcal U= \{U_d: d\in D\}$ of open sets of $X$ such that witnesses that $D$ is closed discrete, ...