# Tagged Questions

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### Homology group of 3-fold sum of projective planes

I want to calculate the homology group of the 3-fold sum of projective planes defined by the labelling scheme $aabbcc$. For this I will use the following corollary from Munkres: Corollary 75.2: Let ...
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### Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
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### An example of a topological space which is a group, but is not a “topological group”

Is there any example of a topological space $X$ which has a group structure, but the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are not continuous?
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### How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...
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### Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
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### 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 ...
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### 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 ...
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### Every metric space has a $(1, 1)$-net

I'm trying to show that every metric space $X$ has a $(1, 1)$-net but struggling - surely $(1, 1)$ is just arbitrary and I've run out of obvious subgroups of $X$ to play with. Any help plz! Here a ...
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### if we set topology on a group like that, is it important?

Let $G$ be a group and $\omega$ be set of all subgroup of $G$. Since $\omega$ is closed under intersection, it is trivial to check that $\omega$ satisfies to conditions to be a base. Thus,Let $T$ be ...
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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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### Is a subgroup of a topological group a topological group?

I'm trying to solve the problem from Munkres: Let H be a subspace of G. (Where G is a topological group). Show that if H is a subgroup of G, then both H and H closure are topological groups. Is it ...
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### How to prove (S1, *) is a topological group?

I'm trying to prove that (S1, *) is a topological group. Where S1 = {z complex : |z| = 1}. So I want to show that S1 x S1 -> S1: (z,w) -> z*w and S1 -> S1 (z) = z^(-1). I'm not really sure where to ...
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### Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
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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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### Infinite Galois Theory

In infinite Galois theory for the one-one correspondence as in the finite case, one needs to introduce Krull topology. What is the intuition behind defining such topology ?
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### Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1}$$are continuous. Is the single requirement that ...
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### Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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### Orbit space of a free, proper G-action principal bundle

Let $G$ be a topological group and let $r \colon E \times G \to E$ be a continuous right-action on a topological space $X$. If $p\colon E \to B$ is a continuous map into a topological space $B$ such ...
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### Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
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### Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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### Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?

On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true? I read it on this page: http://en.wikipedia.org/wiki/Totally_disconnected_group ...
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### HNN extensions as fundamental groups

I have heard that the Seifert–van Kampen theorem allows us to view HNN extensions as fundamental groups of suitably constructed spaces. I can understand the analogous statement for amalgamated free ...
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### Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that \forall a \in G , \forall x \in G ...
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### “Uniform groups” (similar to topological groups)?

Why have I heard about topological groups, but nothing about "uniform groups" (uniform spaces endowed with a group)?