1
vote
1answer
34 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 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
0
votes
0answers
20 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
38 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
18 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
61 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
45 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 $X$. ...
6
votes
3answers
171 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
36 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
63 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
30 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 ...
4
votes
2answers
269 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
69 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
140 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 ...
4
votes
2answers
98 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
106 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 $X$ ...
2
votes
1answer
65 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
123 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
53 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
101 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
192 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
54 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
90 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
65 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
340 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
42 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
90 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
147 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
176 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
60 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
111 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 ...
6
votes
1answer
139 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
124 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
282 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
64 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
127 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. ...
0
votes
1answer
44 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
11
votes
3answers
251 views

Fundamental Theorem of Algebra for fields other than $\Bbb{C}$, or how much does the Fundamental Theorem of Algebra depend on topology and analysis?

When proving the Fundamental Theorem of Algebra, we need to appeal to analytic and/or topological properties of $\Bbb{C}$ and $\Bbb{C}[z]$. Is this going to be necessary in general, and if so, to what ...
9
votes
0answers
186 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
2
votes
1answer
63 views

Cauchy product on topological rings

Let $R$ be any commutative Hausdorff topological ring. I am looking for a preferably general condition on sequences $(x_n)_{n \in \mathbb{N}}$, $(y_n)_{n \in \mathbb{N}}$ such that the equation $$ ...
0
votes
1answer
45 views

Monotonic in sigma algebra

Please help me prove that if $A \subseteq B$, then $m(A) \leq m(B)$ (That $m$ is monotonic). How would you prove this? Can we say $m^*(A \cup B) \leq m^*(A) + m^*(B)$ where $m^*(A \cup B) + m^*(A) = ...
0
votes
3answers
60 views

Lebesgue Measure with given function?

Suppose $E$ subset $R$ ($R $\is real numbers) where $E$ is Lebesgue measurable, and $f:E\to R$ and defined $g: R\to R$ by \begin{equation*} g(x) = \begin{cases} f(x) & x \in E \\ 0 & x ...
6
votes
1answer
73 views

algebraic distance of an element of a ring from an ideal

Let $A$ be a commutative ring and $I$ an ideal. Does there exist a notion of "distance" of an element $x \in A$ from the ideal $I$? This "distance", need not be of the form $A\rightarrow \mathbb{R}$; ...
1
vote
2answers
92 views

Algebraic structure, injection map, standard topology

Suppose that $\varphi : \mathbb{R} \to \mathbb{R}$ is a function which satisfies: $\varphi(1) = 1$ and $$ \varphi(x + y) = \varphi(x) + \varphi(y) , \varphi(x * y) = \varphi(x) * \varphi(y) $$ for ...
10
votes
1answer
227 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
7
votes
1answer
231 views

Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems

There are some set systems with algebraic titles, such as "field", "algebra", "ring" and "semi-ring" (and possibly other titles), in their names. Examples are a sigma field (aka sigma algebra, ...
5
votes
1answer
193 views

When and why do products preserve pushouts?

Let $A,B,C$ topological spaces and then $D$ the pushout of a diagram $$B\stackrel{b}{\leftarrow}A\stackrel{c}{\rightarrow}C.$$ It seems logical to me that for a fifth topological space $E$ the ...
1
vote
0answers
65 views

Definition by commutation property on structures : continuity and where?

(This is very vague, so sorry if there are approximations) I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a ...
16
votes
2answers
661 views

What is algebraic geometry?

I am a second year physics undergrad, loooking to explore some areas of pure mathematics. A word that often pops up on the internet is algebraic geometry. What is this algebraic geometry exactly? ...
3
votes
1answer
77 views

Induced topology on the homomorphic image of a topological group

I would like to do a small sanity check on the following situation: Let $\pi: G \rightarrow G'$ be a surjective homomorphism of topological groups. Let the topology of $G$ be given by a sequence of ...