# Tagged Questions

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### Open and closed equivalence relations

I am looking for canonical examples of open and closed equivalence relations, especially ones that are generated by a continuous functions. Intuitively I think that an open /closed continuous function ...
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### Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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### Irreducible components of $Spec(A)$

A topological space $X$ is called irreducible if given $A_{1}, A_{2}$ open sets $\neq \emptyset$ then $A_{1} \cap A_{2} \neq \emptyset$. The maximal irreducible topological subspaces of $X$ are ...
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### The automorphism group of the real line with standard topology

How much is known about the automorphism group of the real line with the standard topology? I have been unable to find a reference for this question. Any information about $\mathrm{Aut}(\mathbb R)$ or ...
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### Maps from $SO(3)$ to $S^1, S^2$, and $S^1 \times S^2$

I am looking for continuous maps between the special orthogonal group of $3\times 3$ matrices and the unit circle, unit sphere, and their product ($S^1$, $S^2$, $S^1 \times S^2$, respectively). Any ...
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### How does topological dense subgroup induces properties in the larger group?

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
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### Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
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### Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
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### How to show that $\mathbb{Q}_p^*$ is totally disconnected?

Let $\mathbb{Q}_p$ be the field of p-adic numbers and $\mathbb{Q}_p^*$ the set of invertible elements in $\mathbb{Q}_p$. How to show that $\mathbb{Q}_p^*$ is totally disconnected? Thank you very ...
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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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### Categorical introduction to Algebra and Topology

I am self-studying Mathematics in my free time. At the moment I am reading books on Algebra and on Category theory. More exactly, I started working through the book $\textit{Algebra}$ by Serge Lang. I ...
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### Fibre Bundles of Topological Groups

I am trying to construct a variety of manifolds which have a topological group structure. I feel that it is possible to do so by taking two manifolds with group structures and defining a bundle of one ...
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### is $t: \frac{U}{U\cap V}\to \frac{UV}{V}$ not a homeomorphism of topological groups in gerneral?

If U and V are topological subgroups of a topological group G, such that G=UV, and V normal in G, we know that the map $t: \frac{U}{U\cap V}\to \frac{UV}{V}, x(U\cap V)\mapsto xV$ is a isomorphism of ...
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### Is the “vanishing set” the same as the “kernel” of a function?

I've just read of a set called "vanishing set" in my topology lecture notes. It seems to be the kernel of a special type of functions. Definitions The lecture notes are in German, but I try 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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### Can you have an open interval over the set of natural numbers?

Can you have an open interval for instance, over the set of natural numbers? Wouldn't every open interval on the natural numbers simply be another way of writing a closed interval over the naturals? ...
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### Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...
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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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### finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
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### Understanding Pushouts in Top.

The Pushout of $X \leftarrow Z\rightarrow Y$ with $f:Z\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathbf{Top}$ exists and is given by $X\coprod Y/\sim,$ where "$\sim$ is the equivalence relation ...
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### A surjective ring homomorphism $\phi : C([0,1]) \rightarrow \mathbb{R}$ is evaluation at a point [duplicate]

Let $\phi : C([0,1]) \rightarrow \mathbb{R}$ be a surjective ring homomorphism. How would I prove that $\phi$ is the evaluation map $\phi(f) = f(x)$ for some $x \in [0,1]$? I'm not even sure ...
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### Relationship between decompositions of a $G$-variety $V$

Let $V$ be a variety over a field $k$, and let $G$ be an algebraic group over $k$ which acts morphically on $V$. $V$ has three canonical decompositions, and I'm interested in the relationships ...
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### $\mathbb{R}$ - algebras in topological spaces

I´m reading an introduction to $\mathbb{R}$-algebras and in the text there is an observation that says: If $X$ is a topological space, then the set of functions $f : X \to \mathbb{R}$ are a ...
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### Where have you seen this topological ring used?

Let $R$ be a ring and define the topology on $R$ to be that where the open sets are unions of ideals. Then this forms a topological ring - let me know if you need proof. Have you seen it used ...
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### What characterizes topological spaces where every open set is closed?

Motivated by the valuation topology on a discrete valuation ring, which has the above property, I want to know if there's some (subjectively, probably) nicer criterion for when a space has every open ...
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### Differentiable functions on closed and open sets in $\mathbb{C}$

Is there a difference between functions holomorphic (on open sets $\Omega$) and functions that have derivatives everywhere on $\mathcal{Cl}(\Omega)$ (their closure in $\mathbb{C} \cup \{\infty\}$, ...
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### How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
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### Condition for the boolean algebra of clopen sets to be extremely disconnected.

Let $X$ be a topological space and let $\Gamma \mathcal O(X)$ be it's boolean algebra of clopen subsets. For compact totally disconnected space, show that $\Gamma \mathcal O(X)$ is complete (as a ...
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### Metal Ball Cage Template Cardinality: A Brilliantly Lazy PROOF

N.B. - I'm looking for the simplest way to ascertain the number of templates $T$ (see below) comprising the structure from just one angle alone; that is, I'm sitting down looking up at this thing, ...
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### characterizing open subgroups of profinite groups

I am studying Brian Osserman's notes on infinite Galois theory and i am a little bit confused in his proof of Lemma 2.2. In particular, we have a profinite group $G$ being the inverse limit of ...
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### defining a topology on a group by defining a fundamental system of neighborhoods of zero

Let $G$ be an abelian group and $\left\{G_i\right\}$ a family of subgroups. I would like to make sure i fully understand what we mean when we say that "we take the $\left\{G_i\right\}$ to be a ...
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### The Disk and the Punctured Disk

Can you explane me why $$D = \operatorname{Spec}\mathbb{C}[[t]]$$ is the disk and $$D^{\times} = \operatorname{Spec}\mathbb{C}((t))$$ is the punctured disk? Or give me some links on intelligible ...
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### Please list a few topological groups that I should learn about.

I'm going through Munkres' Topology book and there's a lot about topological groups. For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example. ...
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### Continuous group representation

Suppose you have a topological group $G$ , a normed $k$- vector space $V$ and a group homomorphism $\rho:G\longrightarrow GL(V)$. How do you define the topology on $GL(V)$ to make this map ...
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### What mathematical objects permit “taking of limits”?

Background I have been reading a lot of abstract algebra recently (at the level of Artin/Dummit & Foote/Herstein Topics in Algebra for those of you familiar with these books). I have noticed ...
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### The group $\mathbb{Z}^\mathbb{N}/\mathbb{Z}^{(\mathbb{N})}$ can't be embedded in a product $\mathbb{Z}^A$ for any $A$

How the tittle says I need to prove that: There isn't a group monomorphism $\psi: \mathbb{Z}^\mathbb{N}/\mathbb{Z}^{(\mathbb{N})} \to \mathbb{Z}^A$ for any $A$ and, of course, this is equivalent ...
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### Is closure of a semigroup again a semigroup?

Let $S$ be a compact left-topological semi-group (meaning, $S$ is both a semi-group and a compact Hausdorff topological space, and the map $x \mapsto x y$ is continuous for any fixed $y$, but the map ...
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### $G$ is locally compact semitopological group. There exists a neighborhood $U$ of $1$ such that $\overline{UU}$ is compact.

Let $G$ be a group and $\mathcal T$ be a locally compact topology on $G$ such that for any $a\in G$ the functions $x \mapsto ax$ and $x\mapsto xa$ are continuous on $G$. How to prove elementarily ...
### $[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ provided $T \circ T = T$ [closed]
Let $V$ be a finite-dimensional vector space and let $T:V \rightarrow V$ be a linear map such that $T \circ T = T$. How should one prove that there is a basis $\beta$ of $V$ such that \begin{eqnarray} ...
### Orthogonal group is a regular submanifold of $GL(n,\Bbb R)$
I want to show that $O(n)$ is a regular submanifold of $GL(n,\Bbb R)$. I think that I can use constant rank theorem but how? I am putting the picture that what I did. Please help me I want to learn. ...