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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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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62 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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94 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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215 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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117 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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90 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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132 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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209 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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163 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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177 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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168 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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434 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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63 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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106 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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82 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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55 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 ...
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55 views

defining relationship between geometric entities (features)

I have different features located on a plane (2D); I want to define this structure mathematically in a way to represent their relations. Some of features are aligned in horizontally, vertically or ...
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326 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then $Y_0\...
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206 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
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162 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
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224 views

Topological proof that this set is a topological manifold

let $S \subseteq \mathbb R^3 \times \mathbb R^3$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show that this is a topological manifold without ...
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178 views

Find the closure for several sets

(a) $\mathbb{Q}$ (b) {$(x,y)\in\mathbb{R}^2:xy<1$} (c) {$(x,\sin($${1}\over{x}$$)):x>0$} (d) {$(x,y)\in\mathbb{Q}^2:x^2+y^2<1$} First Closure $\overline{A}$, it is a set contains all ...
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44 views

Topological inequivalent manifolds obtaining by removing a surface from a manifold

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
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76 views

Prove that you can't connect both pairs of opposite sides of a square without the two paths intersected.

Formally, let $$D=[-1,1;-1,1]\subset\mathbb{R}^2,$$ and let $f,g:[0,1]\to D$ be two continuous functions, such that $f(0)=(-1,0)$, $f(1)=(1,0)$, $g(0)=(0,-1)$, $g(1)=(0,1)$. Prove that $\exists\zeta,\...
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24 views

How to build the space BTOP

Can anybody explain how is the procedure for building the space BTOP, which classifies microbundles of topological manifold ? Is there any good (and easy to read) references on this subject ? Thanks ...
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88 views

How to show that for any meager set $A$ in Baire Space, there is a nowhere dense set $C$, such that $A \subseteq C^*$?

Let $X$ be a subset of $\omega^{\omega}$, $X^{*}$ is defined as:$$\{y:(\exists x\in X,\exists N <\omega)(\forall n >N x(n)=y(n))\}$$ which consists of all sequences in $\omega^{\omega}$ that ...
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307 views

Different definitions of regularity of a measure

I was wondering what relations are between these different definitions of a regular measure? When are they equivalent? There are two non-equivalent definitions from Wikipedia Let $(X, T)$ be a ...
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192 views

sub-basis for a topology on the real line

Consider the closed intervals $[a,b]$ as a sub-basis for a topology on the real line. Describe the resulting topology My attempt If $[a,b]$ is open, and $[a-1,a]$ is open, then the intersection of ...
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84 views

Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
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64 views

Inverse limits and sums

Let $(X_\alpha)_{\alpha<\omega_1}$ be a family of compact metric spaces such that $X_\alpha$ is homeomorphic to a subspace of $X_\beta$ for $\alpha<\beta$. Can we regard the disjoint sum $\...