Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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394 views

Recovering the topology of an affine scheme from the specialization preorder

Let $A$ be a commutative ring. The specialization preorder on $\mathrm{Spec}(R)$ is given by $\mathfrak{p} \prec \mathfrak{q} \Leftrightarrow \mathfrak{p} \in \overline{\{\mathfrak{q}\}} ...
2
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69 views

Unit partition to produce smooth function from continuous ones

Given a positive continuous function (except on closed set, where is zero ) on a smooth manifold how to find a smooth function under the same conditions being less (or equal) than this one ...
2
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0answers
73 views

$\sigma$-product space and $ \Sigma$-product space

Recently, I'm interested in the $\sigma$-product space and $\Sigma$-product space. Is there a survey on $\sigma$-product space and $\Sigma$-product space, which is simple for a beginner? Thanks for ...
2
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148 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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191 views

Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
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245 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
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111 views

Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho ...
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922 views

Are the continuous functions pointwise dense in the bounded measurable functions

Suppose we have a compact set $K$. I know that the space $C(K,\mathbb{C})$ of continuous functions is complete with respect to the norm $\|f\| = \sup_{x\in K} |f(x)|$. Let $L^{\infty}$ be the space of ...
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88 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
2
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89 views

Total sets in $R$ compared with total sets in $[a,b]$

A total set in a NLS is one whose linear span is dense in the set. e.g. $A = \{1,x, x^2,...\}$ is total in $(C[a,b],\Vert\cdot\Vert_{\infty})$ I find it easier to talk about total sets than dense ...
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411 views

The topology of distributions

I have been wondering about the following concerning the spaces $\mathcal D$ of test functions (say on $\mathbf R$). It is my understanding that the topology on this space is inductive limit ...
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607 views

Topology of wedge products

I have a question about the quotient topology induced on the wedge sum $S^{\,2} \vee S^1$, (where $S^n$ denotes the unit sphere in $\mathbb{R}^n$). In this topological space, the subsets $S^1$ and ...
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112 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
2
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286 views

Relations and differences between outer/inner limit and Kuratowski limsup/liminf

Let $X$ be a topological space. I am asking about the relations and differences between the following two different types of $\limsup$ and $\liminf$ of $A_n ⊆ X, n ∈ \mathbb{N}$, a sequence of ...
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354 views

Vector calculus- vector field and path

Let $U\subset \mathbb{R}^2$ be open, and $F:U\to \mathbb{R}^2$ a $C^1$-vector field. Assume that: $$\frac{\partial{F_1}}{\partial{x_2}}(x)=\frac{\partial{F_2}}{\partial{x_1}}(x)\quad\forall x\in U$$ ...
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96 views

factor of covering map is a covering map?

A paper I'm trying to understand uses the following lemma: Let $p: U \to U_0$ be a topological covering map. Suppose that we can write $p =\pi \circ f$, where $f:U \to Y$ is an open surjective map, ...
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94 views

topology exercise. compactness circle projective space.

Is the circle compact in $\mathbb{P}_{2}(\mathbb{C})$? Here what I did: I considered the circle in $\mathbb{C}^2$ is $\{(x,y)\in\mathbb{C}^2|x^2+y^2=1\}$. The projective closure in ...
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165 views

Trying to prove a intuitively “obvious” fact.

I'm trying to prove that all continuous maps of pairs $f:([-1,1], \{-1,1\})\to (\{-1,1\},\{-1,1\})$ are constant, and I've almost got a working argument, but it reduces down to the following ...
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81 views

Does it imply a lift?

Let $p:S^1\times S^3\rightarrow S^1\times S^3$ be a covering map with $p(z,y)=(z^3,y)$ and $z\in S^1\subset\mathbb{C}$ and $h:\mathbb{R}P^4\rightarrow S^1\times S^3$. Is there a lift ...
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210 views

Why a spiral is the deformation retract of a plane?

As the title says, why a spiral is the deformation retract of a plane?.
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324 views

Do we need net refinements not induced by preorder morphisms?

From Engelking's book on general topology (slightly rephrased): Definition: We say that the net $S': \Sigma' \to X$ is finer than the net $S: \Sigma \to X$ if 1. there exists a function $f: ...
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197 views

homework problem about the projective real space

Sorry for ask this problem, but I am very complicated with this problem :/ . My course it´s of topology, the teacher said that we only need the definition of the quotient topology and of $$ P_R^2 ...
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47 views

Set of 3D surfaces

How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite? I am guessing that we could either use proof by contradiction or come up ...
2
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279 views

Point-set topology and set theory

In the standard second-order, but single-sorted setting of point-set topology one has a base set $X$ and the property of being open on its powerset $P$ obeying the usual axioms. Proofs in point-set ...
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273 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...
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467 views

Properties of the universal cover of CW-complexes

Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact: $X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
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376 views

Learning analysis through topology

One of my supervisors once mentioned that when he was learning analysis he learnt it backwards. He learnt topology first and then saw analysis after, instead of the usual approach of doing everything ...
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292 views

Direct proof to show that a set is closed

I was trying to prove something, and I did it, but what I used is too exaggerated. The problem is: Let K be the cantor set, prove that the sets $$ \eqalign{ & \left\{ {\left| {x - y} ...
2
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266 views

uniform distribution on unit ball

If $S$ is a set of an countably infinite number of points uniformly distributed throughout the unit ball in $\mathbb R^n$, is there for every point $p$ in the ball and every real number $e>0$, a ...
2
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133 views

Property (T) for groups vs top

I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you. ...
2
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110 views

Contractions and Map Extensions

I'm going through Spanier and got stuck on the following problem: Show that a space $Y$ is contractible if and only if given a pair $(X,A)$ having the homotopy extension property with respect to $Y$, ...
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677 views

homeomorphisms mapping interiors to interiors and boundaries to boundaries

Why do homeomorphisms map interiors to interiors and boundaries to boundaries? I cannot find a good proof for it that does not involve algebraic topology. I only need it for spaces in ...
2
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192 views

how to factor a map by a group action

Let $X$ and $Y$ be topological spaces and a surjective map $f:X\rightarrow Y $. Suppose that a group $G$ acts on $X$. and let $\pi:X\rightarrow X/G$ be the quotient map. 1) Under what conditions $f$ ...
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505 views

How do you prove a CW complex is locally path connected

I think this is done inductively on the skeletons but I can't work out the details.
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302 views

fundamental group of closed surfaces as CW complexes

Let $T$ denote the $2$-torus, $P$ the projective plane, and $nT$/$nP$ the connected sum of $n$ tori/$n$ projective planes respectively. 1) how can I prove, that $nT$ and $nP$ are homeomorphic to ...
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146 views

One special set on [0,1]

Let $X = [0,1]$. Define $f:X\to\mathbb{R}_{\geq 0}$ to be Lipschitz continuous on $X$. Put $$Y\subset X:\int\limits_Y f(x)\,dx = 0$$ What can we say then about $A = X\setminus Y$? It is not defined ...
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84 views

Questions about boundary faces of simplices and triangulations

Let $S$ be a simplex in $\mathbf R^n$ and let $\{S_i\}$ be a triangulation of $S$. The boundary of $S$ is defined as the union of the boundary faces of $S$. Is this union equal to the topological ...
2
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148 views

How to describe all continuous maps from T to T'?

How to describe all continuous maps from $T'$ to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and $T'=\mathbb{R}$ with the topology with basis given ...
2
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190 views

On the Compact Uniformization Theorem

I just read through a proof of the Compact Uniformization Theorem, and I follow it up to the very last line. The proof is: Compact Uniformization Threorem. If $X$ is a compact regular space, then the ...
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0answers
37 views

Why does excision imply this?

In exercise $4$, page 230 of Bredon, he asks for a proof of the Mayer-Vietoris sequence using a commutative braid diagram which substitutes some terms by others using excision. I've solved the ...
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0answers
28 views

Order topology on $\Bbb Q^+$ as a Z-module

Consider the multiplicative group of strictly positive rationals, denoted $\Bbb Q^+$. This can be viewed as a $\Bbb Z$-module, with the primes serving as a basis. If we place the order topology on ...
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17 views

Neighbourhoods with proper multiplication

Assume we have two closed subsets $F$ and $G$ of $\mathbb{C}^*$ which are proper for the multiplication, i.e. $$KF^{-1}\cap G$$ is a compact of $\mathbb{C}^*$ when $K$ is a compact of $\mathbb{C}^*$. ...
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0answers
57 views

Impossible Covering Properties for Sets of Reals

I've been reading more about selection principles (covering properties) recently. Below is terminology. Adapting what B. Tsaban said in this article, we consider spaces $X$ which are (homeomorphic ...
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28 views

An example of infinite Betti number refers to G-covering, and the motivation of G-covering

I saw from a book that if G is infinite, then when considering G-covering: $p:X'\to X$, the P-th Betti number of $X'$ can be infinite. Can you give me an example of that. Also I am curious why we need ...
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35 views

Example of two affine varieties $X,Y$ such that the image of $\phi:X \rightarrow Y$ is not locally closed

In my course Algebraic Geometry I always find it hard to come up with examples or counterexamples. For instance in the following question: Give an example of two affine varieties $X,Y$ and a morphism ...
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21 views

Connected sum $S_1$ # $S_2$ is commutative and associative

The connected sum of two surfaces $S_1$ and $S_2$ is formed by removing a circular hole from each surface and identifying the boundaries together Show that the connected sum $S_1$ # $S_2$ is ...
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0answers
13 views

Show that $C/B \approx T/A$

Let $T=S^1 \times S^1$ be the torus with meridian $A=S^1 \times \{1\} \subset T$ Let $C=S^1 \times [-1, 1]$ be the cylinder with base circles $B=S^1 \times \{-1, 1\}$. Show that $C/B \approx ...
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0answers
13 views

existence of quotient maps to topologically transform one shape into another

What are the mathematical condition that are necessary for a shape A to be transformed into shape B by quotient maps ? For example , I can divide a square into two triangles by drawing a diagonal .Now ...
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0answers
25 views

Stone-Cech compactification and maximal ideal of C(X)

Please i want to know the usefulness of the "Gelfand and Kolmogoroff theorem" when showing that $\beta X$ of a a completely regular Hausdorff space $X$ Can be identified with the Structure space of ...
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24 views

Rewording the definition of closure

In Munkres there was a statement: Given a topological space $(X, \tau)$ $x \in \overline A \iff \text{ for every open set } U \text{ containing } x, U \cap A \neq \varnothing$ Following from ...