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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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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91 views

Product varieties with the constructible topology

Let $k$ be an algebraically closed field and let $X\subseteq k^n$, $Y\subseteq k^m$ be two affine algebraic varieties. It is not difficult to find examples where the Zariski topology on the product ...
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85 views

From Jordan's Curve Theorem to Jordan-Schoenfliess theorem

I am trying to learn and understand proofs of classical theorems and successfully mastered a proof of JCT. (It was the well-known proof that uses Tietze Extension and Brouwer's fixed point theorem). ...
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118 views

Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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189 views

Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
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306 views

Construction of Lakes of Wada

At each step of the construction of Lakes of Wada we extend a lake (an open set in the open unit square) so that no point of the land (the complement of all the lakes) is farther than a given small ...
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58 views

Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
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59 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
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46 views

Is there a name for a set where any two elements are separated by a given distance?

I am curious if there is a name for such a set. Let $(M,d)$ be a metric space and $S$ a subset of $M$ for which there is some positive number $\delta$ such that for any two distinct elements in $S$, $...
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238 views

An exercise from the handbook of set-theoretic topology

This is an exercise from the handbook of set-theoretic topology (Exercise 13.3): Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact ...
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80 views

derivatives using epsilon-delta argument

I have this question below and am not sure if my approach is correct. Can anyone please advise me? Thanks. Question: Let $f:\mathbb{R}^n \to \mathbb{R}^m$ and suppose there is a positive constant $K$ ...
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58 views

Finding a sufficient condition for a set to be finitely decomposable into open sets..

Let us call a set in a topological space finitely decomposable set (FDS) iff it can be rewritten using the standard set operations $\cup$ and $\sim$ and only finitely many open sets. I'm looking for ...
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124 views

Pushing a map off a disk

Let us assume that we have covered $\mathbb{R}^n$ with the open sets $V = 2 \cdot int(D^n)$ (the standard unit disk, and 2 means multiply the size by 2) and the family $\mathcal{U}$ of open disks W ...
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34 views

Let $(X,\tau)$ be a $T_B$-space…

A topological space is called $T_B$ if every compact subset is closed. (I):$Let (X,\tau)$ be a $T_B$-space which is not countably compact, $\{x_n :n \in \omega\}$ a set without accumulation points, $\...
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89 views

Compact Hausdorff implies product of quotient map is a quotient map?

Let $X$ be compact Hausdorff and let $q : X \to Y$ be a quotient map. Is it true that $f : X \times X \to Y \times Y$ with $(x_1, x_2) \mapsto (q(x_1), q(x_2))$ is a quotient map?
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65 views

Isotopy between two open disks on a surface

So I have a (compact) surface $\Sigma$ and two open disks on the surface call them $A$ and $A'$ such that the intersection contains a simple curve $P$. What I want to do is construct an isotopy ...
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68 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
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97 views

Counter-example about paracompactness

I am trying to find a counter-example related to the definition of paracompactness, but it seems that it is not very easy. Here is the problem. Give an example to show that if $X$ is paracompact, ...
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219 views

Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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118 views

Hartshorne II Prop 2.6

Prop 2.6 constructed a continuous map $X$ to $t(X)$, I cannot verify that it is a homeomorphism. I try to show any open set $U$ is mapped to $t(X)\setminus t(X\setminus U)$. To show it is surjective, ...
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49 views

Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
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91 views

Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
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55 views

Not 1-dimensional homological equivalent of the circle

The questions origins from this problem and my incorrect answer to it. I'm trying to correct it, but it turned out that the topological space - that I need to do it straightforward - has very specific ...
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73 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
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84 views

Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
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47 views

Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
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133 views

Union of Sets in Locally Compact Hausdorff Space

Is it possible for an open set in a locally compact Hausdorff space to not be the union of an increasing sequence of compact sets? If so, given a regular Borel measure on such a space, how is it that ...
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210 views

Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
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70 views

intro. to topology mendelson - closure in a subspace

I'm self studying intro to topology by Mendelson and I just completed a book problem and wanted to get input on whether it's okay. The problem statement is, Let $Y$ be a subspace of $X$ and $A\...
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165 views

Covering argument

In proving Harnak's inequality (I am referring to this article: "On Harnack’s Theorem for Elliptic Differential Equations"Communications on Pure and Applied Mathematics Volume 14, Issue 3 ), Moser ...
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143 views

How to turn a topological space into a semi-decidable logic?

In two interesting posts(here and here),it is mentioned that "there is a close connection between semi-decidable logics and topological spaces" Michael O’Connor wrote: In fact, given a ...
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184 views

Isolated points

Every point $x \in S \subset\Bbb R$ is isolated. 1) $S$ is closed? 2) $S$ doesn't have any limit point? My attempt: by definition any isolated point is boundary point, and cannot be limit point. ...
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1k views

Showing the Unit Circle is Connected

One way to show that the unit circle is connected is to use the map $f: [0, 2\pi] \to \mathbb{R}^2$ where $f(x) = (\cos x, \sin x)$. Since $f$ is a continuous map and $[0, 2\pi]$ is connected, the ...
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171 views

Intersection form on manifolds with boundary

It is a "basic fact" that the intersection form of a closed oriented 4k-dimensional manifold is unimodular. (Could anyone point me to a reference to a proof of this fact?) What can be said about the ...
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439 views

If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact

Let $X$ be a Banach space and $C\subset X$. $\fbox{1}$ If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact ? $\fbox{2}$ If $C$ is convex , weakly-closed $\...
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53 views

Can you construct a coutable local base in the space of continuous functions?

Let $(C,\tau)$ be the topological vector space of all complex continuous functions on $[0,1]$ with seminorms $p_x(f)=|f(x)|$, $x\in [0,1]$. We have known $(C,\tau)$ is not metrizable,but how could I ...
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55 views

Topological graphs

Given the universel covering space $\hat{X}$ of $X$ by $p:\hat{X}\rightarrow X$, there exists a bijection between subgroups $H<G=\pi_1(X,x_0)$ and covering spaces $\tilde{X}\rightarrow X$ with $\...
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54 views

field lines terminating at infinity

A dipole consists of two equal and opposite point charges separated by a fixed distance. With two exceptions, all the electric field lines begin on one charge and end on the other. In the two ...
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112 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
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66 views

A question on semi-stratifiable spaces

A space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: (i) for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ...
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53 views

Jordan curves, its interiors and the existence of a continuous function.

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ is a Jordan curve, $O(t)$ is its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, is there exists a continuous ...
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160 views

a problem on metric spaces

I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let $(X, d)$ be a metric space ...
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107 views

Compute $df_1: ST_1^3 \rightarrow TSO(3)_I$

In short, the problem is to compute $df_1: T_1S^3 \rightarrow T_{I}SO(3)$, given $f: S^3 \rightarrow SO(3), r \in S^3, f(r) \in SO(3): f(r)(q) = rqr^{-1}, q\in R^3$. I just get to study differential ...
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135 views

Problems about continuity of $|f|$ and $f\vee g$; confusion about definitions

I can't seem to wrap my head around this notation of my textbook can some please explain to me what this says? What I am trying to show? (a) Given $f: D \to \mathbb {R}$, let $|f|$ be the ...
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89 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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112 views

Cantor set as a set of continued fractions?

Does the classical cantor set have a nice description as a set of continued fractions? I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ...
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211 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
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83 views

A question on star $\sigma$-compact spaces

A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\...
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56 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. $X$...
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77 views

Very Simple Universal Covering problem

A space $X$ is constructed from two disjoint copies of $RP^3$ and a copy of the unit interval $I$ by gluing one end of $I$ to a point of one copy of $RP^3$, and gluing the other end of $I$ to the ...