0
votes
1answer
28 views

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 ...
1
vote
1answer
51 views

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 ...
0
votes
2answers
64 views

A question about the Zariski Topology

Let $\{a_i\}$ be an infinite set of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty a_i$ not defined? I am trying to understand Zariski Topology. Here, $V(\bigcap a_i)= ...
1
vote
1answer
52 views

Residually Finite group $\Rightarrow$ Totally disconnected

How can I prove that a residually finite group $G$ is totally disconnected? I considered the topology generatad by $\{Ng\}_{N\in\eta,\;g\in G}$ where $\eta=\{N\unlhd G \;, |G:N|<\infty\}$ and I ...
1
vote
1answer
51 views

Simple question on topological groups

Why is $\{1\}$ closed in a totally disconnected topological group?
0
votes
2answers
98 views

Difference between Topological Data Analysis and Graph Technology

I'm trying to understand the difference between Oracle's graph technology which apparently has an inherent understanding of topology and Ayasdi's Topological Data Analysis technology. Are these two ...
2
votes
2answers
50 views

About Path-connected

Let $a=(a_1,a_2,...,a_k)$ and $b=(b_1,b_2,...,b_k)$ be points in k-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in ...
6
votes
1answer
84 views

Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
1
vote
1answer
63 views

Showing that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$

I have a question that asks me to show that $\mathbb{R}\times [0,1]$ is quasi-isometric to $\mathbb{R}$ I have having trouble showing what I have is a quasi isometry. My map is simply: ...
2
votes
2answers
71 views

Closure of a topological group

Let $G$ be a topological group. Define $H(g)=\{g^n\}^{\infty}_{n=-\infty}$ for each $g \in G$. I need to prove that the closure of the set $H(g)$ is a commutative subgroup of $G$. But I am not sure ...
6
votes
1answer
61 views

Baumslag–Solitar $B(1,2)$ is not hyperbolic

I have a question which asks me to show that the Baumslag–Solitar $B(1,2)$ is not hyperbolic by considering its Cayley graph and showing that triangles can be arbitrarily fat. The Cayley graph can be ...
3
votes
2answers
84 views

Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
2
votes
2answers
73 views

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 ...
10
votes
4answers
293 views

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 ...
1
vote
1answer
59 views

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 ...
0
votes
0answers
23 views

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 $z \rightarrow \alpha z + \beta$ ...
1
vote
2answers
43 views

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 $z \rightarrow \alpha z + \beta$ ...
1
vote
0answers
23 views

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 ...
1
vote
1answer
66 views

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 ...
2
votes
0answers
51 views

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 $X$. ...
7
votes
3answers
227 views

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 ...
2
votes
1answer
43 views

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 ...
1
vote
0answers
64 views

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 ...
1
vote
1answer
34 views

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 ...
3
votes
2answers
500 views

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 ...
1
vote
1answer
94 views

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? ...
6
votes
0answers
150 views

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 ...
5
votes
2answers
119 views

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 ?
2
votes
3answers
107 views

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 $X$ ...
1
vote
1answer
72 views

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 ...
2
votes
1answer
148 views

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 ...
2
votes
1answer
56 views

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 ...
5
votes
1answer
49 views

$\mathbb{R}$ - 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 ...
2
votes
1answer
104 views

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 ...
8
votes
3answers
217 views

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 ...
0
votes
1answer
57 views

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 $\mathbb{C} \cup \{\infty\}$, ...
4
votes
0answers
108 views

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 ...
0
votes
1answer
77 views

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 ...
13
votes
3answers
347 views

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, ...
1
vote
2answers
44 views

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 ...
0
votes
1answer
99 views

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 ...
3
votes
1answer
183 views

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 ...
5
votes
3answers
182 views

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. ...
1
vote
2answers
62 views

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 ...
4
votes
3answers
112 views

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 ...
7
votes
1answer
148 views

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 ...
2
votes
1answer
141 views

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 ...
5
votes
1answer
289 views

$G$ 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 ...
2
votes
1answer
65 views

$[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ 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} ...
2
votes
1answer
146 views

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. ...