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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268 views

interior, boundary, closure of some sets in plane

$\newcommand{\intr}{\operatorname{int}}$ For the following sets $E ⊆ \mathbb{R}^2 $, I need to find $E'$ , $\overline E$, $\intr(E)$ and $∂E$. (a) $E = \{(x, y) : 1 < x^2 + y ^2 ≤ 4\}$, (b) $E = ...
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34 views

Term for Sets Equivalent Up To Closure

Is there a nice name for the collection of sets that have the same closure? I'm writing up some notes developing some tricks for visualizing topologies, and I kind of want a non-cumbersome way to ...
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37 views

If $f, g: S^1\to \mathbb C$ are two functions, what is a homotopy from $f=\frac{g}{\vert g\vert}$ to $g$?

If $f, g: S^1\to \mathbb C$ are two functions, what is a homotopy from $f=\frac{g}{\vert g\vert}$ to $g$? I just want to check whether my homotopy $H(x,t): (1-t)f+tg$ where $x \in S^1, t \in [0,1]$ ...
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46 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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70 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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132 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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63 views

Maps with inverses that preserve dense sets.

Let $\mathcal A$ and $\mathcal B$ be topological spaces I want to consider the set of maps $f:\mathcal A\to\mathcal B$ such that if $S\subset \mathcal B$ is dense in $\mathcal B$ then the preimage ...
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26 views

Calculating Topological Genus

How would I calculate the topology of a sphere, with a smaller sphere inside removed. I know if I drill through to get to the hole, then we are back to being a sphere, and if I drill out to the other ...
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54 views

How to see that $D(\mathscr B)= E(\mathscr B)$ if, in addition, $X$ is locally compact?

Let $X$ be a Tychonoff space. Let $\mathscr B$ be an open base, consisting of clopen sets, for a space $X$. Put $$D(\mathscr B)= \{S=\langle S_n: n\in \omega\rangle: S \text{ is a sequence of disjoint ...
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35 views

Results showing that convexity implies continuity

What results can be used to prove upper semi continuity of the infimum of a convex function? This is in context of duality, topological spaces, hypographs.
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65 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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31 views

Not every $1$-nullset is an $s$-nullset, for $0 < s < 1$

For any positive real number $s$, define an $s$-nullset as a subset $A$ of the real line such that, for any $\epsilon > 0$, there exists a sequence of intervals $\{I_n\}_{n=1}^{\infty}$ having the ...
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47 views

Questions on the product of the space which has countable extent

This question can be seen the continuation of this question and this question. First, let us recall some relevant definitions. Definition 1: A space $X$ has countable extent if every uncountable ...
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52 views

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.

Let $T$ denote the torus $S^1\times S^1$ and let $M$ denote the Möbius band. Let $C$ be a simple closed curve in $T$ which bounds a 2-disk. Form a space $X$ by identifying the boundary of $M$ with $C$ ...
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143 views

Find spectrum of the operator

I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem. Let $\theta \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator $Au=v,$ ...
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80 views

Verifying two proofs about connected sets

Let $A_n$ be a sequence of connected sets such that $A_n \cap A_{n+1}\neq \emptyset$. Show that $\cup A_n$ is connected. Let $A=\cup A_n$. Suppose $A$ is separable and let $U,V$ be a separation ...
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139 views

Show that an induced map on a quotient space is a homeomorphism.

Consider the unit square $Q := [0,1] \times [0,1]$ with the coordinates $(x,y) \in Q$. Let $Z := Q/\thicksim$ be the quotient space, which results by identifying: \begin{align*} (x, 0) & ...
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157 views

Application of Stone Weierstrass Theorem for trigonometric polynomials

In the space $C[-\pi, \pi]$ equipped with the sup norm, consider the linear space $M$ spanned by the functions ${ (\cos nx)}_{n\geq0}$ and $ {(\sin nx)}_{n\geq1}$. what is the closure $\bar{M}$ of ...
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36 views

Is the space $X$ in the class dual to the spaces with the Souslin property?

Recall that $X$ is in the class dual to the spaces with the Souslin property: For any neighbourhood assignment $\{O_x: x\in X\}$, there is a subspace $Y \subseteq X$ such that $c(Y)=\omega$ and ...
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129 views

The Lebesgue number property and uniform continuity (proof check)

Theorem If $f$ is continuous on a compact metric space $X$, then $f$ is uniformly continuous on $X$. Proof Let $\epsilon>0$. For any $y\in X$ there is a $\delta_y$ such that $d(x,y)<\delta_y$ ...
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146 views

The Cantor Space and open, but not closed sets.

consider the space $\{0,1\}^{\mathbb{N}}$ of all infinite binary sequences, called the Cantor-Space. This space is metrizable with metric $$ d(u,v) = 2^{-(r-1)} \qquad \textrm{ where } r = ...
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43 views

All finite Baire measures are Closed-regular?

Given a finite Baire measure $\mu$ on a topological space $X$, is it true that $\mu$ is closed-regular? Where closed regular means that, $$\mu(A) = \sup\{\mu(K)| K \space \text{is a } Z\text{-set}, ...
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25 views

Covering space problem from an old Qual

Suppose that S$^1 \times P^2$ covers some space, and let $h$ be a covering translation. Show that the induced isomorphism $h$ of $H_1(S^1, P^2)$ must be the identity
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44 views

How to build the largest sambusa

I was making sambusa last night. Typically when mama cooks them she has small circles of dough, but mama is not here so I went to the store and bought the dough, and it came in the shape of a ...
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131 views

An example of a differentiable manifold class $C^k$ but not class $C^{k +1} $

I'm looking for an example of a differentiable manifold of class $C^k$ but not class $C^{k +1}.$ I found an exercise in Hirsh's book, which suggests that the graph of $f (x) = |x|^{\lambda}$, where ...
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66 views

How to deduce the results of response time by this trajectory approach?

First, we denote this: And And we get this right property( $last_i$ means the last node on $τ_i$): And: $Smin_i^h$ = $\sum_{h'=first_i}^{h-1} ({C_i^{h'} + L_{max})}$ $Smax_i^h$ = ...
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56 views

Example of function which are closed function but may not be continuous

I've found that function both open and closed but fail to be continuous. Now, I want to find an example of closed mapping that may not be continuous.
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133 views

Proving that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$

I'm trying to prove that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$. The intermediate step is proving that: if $a$ is a lift of $f \circ \exp$ and $b$ is a lift of $g \circ \exp$ then $a+b$ ...
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73 views

Smoothing corners of a handle attachment

Say we attach a $\lambda$-handle, $\mathbb{D}^\lambda \times\mathbb{D}^{\mu}$, to a smooth manifold $M, \partial M$ by simply taking the quotient $M \cup_h \mathbb{D}^\lambda \times\mathbb{D}^{\mu}$ ...
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83 views

Question on homotopy lifting

I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear. A lemma in my lecture notes says: Let $p: \tilde Y \to ...
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89 views

Orientation of the barycentric subdivision

Two orderings of the vertices of an $n$-simplex are said to be equivalent if they differ by an even permutation. An orientation of an $n$-simplex is a choice of one of the two equivalence classes of ...
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65 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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128 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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76 views

Continuity and openness of the map $C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$

I need to prove or disprove that the composition operator is continuous and open. Consider the following map $$h:C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$$ that takes a function ...
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66 views

Completely Regular Topological Space and Measure Theory

Here is the statement... Suppose that $(X,\tau)$ is a comletely regular topological (I think the lecturer requires X to be Hausdorff too.), and that E is a dense linear subspace of ...
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555 views

Collection of open intervals $(a,b) \ a,b\in \mathbb{Q}$ is a basis for euclidean topology on $\mathbb{R}$

I'm not sure if this question hasn't already been asked here, but I couldn't find it. I'm currently studying topology and I'm reading a book which unfortunately has no answers to the exercises. ...
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36 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
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45 views

Sierpinski's triangle set of local cut points

Could you please help me identify Sierpinski's triangle set of local cut points? I know it does not have cut points but it has local cut points.Which are they? Thank's in advance !
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33 views

Polar representation of continuous curve.

Given a continuous curve $\gamma:[a,b]\to\Bbb R^2$ with $\gamma(t)\ne(0,0)^T$, can I always find continous functions $\rho:[a,b]\to\Bbb [0,\infty)$ and $\theta:[a,b]\to\Bbb R$ so that ...
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87 views

Questions on compact space

Let $X$ be compact. Q1, Does there exist a dense subset $Y$ of $X$ such that the tightness of $Y$ is countable? Q2, Does there exist a dense countably compact left-separated subset $Y$ of $X$? ...
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92 views

Analysis and real analysis

Let $X$ and $Y$ be compact metric spaces. Let $X\times Y = \{(x; y) \,:\, x \in X;\, y \in Y \} $be the cartesian product. Show that any $f \in C(X \times Y )$ can be uniformly approximated by ...
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72 views

Why are pointwise and uniform convergences topologizable?

My question is about two types of convergence on functional spaces. Why can pointwise convergence for mappings from a set to a topological space be topologizable? Why can uniform convergence for ...
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97 views

Continuity in the Strong(Whitney) Topology

Let $P,M$ and $N$ are smooth manifolds and let $F:P\times M \to N$ be a smooth map. We know its associated map $\tilde F:P \to C^\infty(M,N)$ given by $p \mapsto F_p(m)$ is continuous if and only if ...
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111 views

2-sphere union with unit 2-cell is homotopy equivalent to one point union of two 2-spheres.

This is problem 2 in Bredon I.14, on homotopy. I need to prove that $X$ = union of the 2-sphere with the unit 2-cell going through the origin is homotopy equivalent to $Y$ = one-point union of two ...
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144 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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67 views

Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
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670 views

Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
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102 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
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93 views

Two problems on topological properties of two different sets

Which properties hold for the following sets? Open, connected, closed, nowhere dense? $$A =\{(x,y) \in\mathbb{R}^2 \mid y=mx\}\setminus \{(0,0)\}\subset\mathbb{R}^2$$ $A$ is the ...
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85 views

A theorem of a continuous map $f: S^1 \to S^1$

This is a theorem from my lecture notes: If the continuous map $f: S^1 \to S^1$ extends to a continuous map $F: B(0,1) \to S^1$ the $f$ is homotopic to a constant map. The proof just defines a ...