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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405 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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600 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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283 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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95 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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209 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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272 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 ...
2
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463 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 $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
2
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375 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. ...
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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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667 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 ...
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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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500 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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300 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 ...
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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 ...
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189 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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17 views

Prove that the union of two given subsets of $\Bbb{C}^n$ is path-connected

Consider a subset $A$ of $Z=(\Bbb{C}^n$, Zariski topology) and regard it as a subspace of ($\Bbb{C}^n$, Metric topology). Sine $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^2n$, we can decide if A is ...
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24 views

Does a certain type of connected subset exist in Euclidean spaces having an arbitrarily high dimension

Let $\mathbb R$ be the set of all real numbers and for each positive integer $n$, let $f_n$ be a mapping of $\mathbb R^n$ into $\mathbb R$. For each positive integer $n$, does there exist an $f_n$ ...
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42 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
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7 views

Definition of symmetry of the open ball in a normed space.

I need to prove that every ball in a normed space is a convex set symmetric with respect to the center of the ball. I have problem with defining symmetry. Let $X$ be a normed space. And $B_r(a) ...
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25 views

Quotient map on Real line

Let $X=\Bbb R$ and $q: X \rightarrow(\Bbb R, T_{std})$ be a function $q(x)= \begin{cases} 1-|x|, & \text{x $\le1$} \\ |x-1|, & \text{x $\ge$1} \end{cases}$ Q1 Suppose X is given the ...
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29 views

Show spaces are not homotopy equivalent.

Show that the spaces $X=S^2xS^2$ and $Y=S^2\vee S^2\vee S^4$ are not homotopy equivalent. I read somewhere that: We have that $\pi _1(X),\pi _1(Y)$ are trivial and that $X,Y$ have isomorphic ...
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19 views

$star(v)$ for a 0 simplex

Suppose you have a simplicial complex and a vertex $v$ which is not connected to any other vertex. Is $st(v)$ just the empty set? If you're looking at the inside of a simplex you don't look at ...
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38 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
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18 views

Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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43 views

Does this function have a dense graph?

Let $\mathbb Q =\{q_n:n\in\mathbb N\}$ be an enumeration of the rationals. Let $f(x)=\mid\sin(1/x)\mid$ if $x\neq 0$ and $f(0)=0$. Let $g(x)=\sum_{n=1} ^\infty \frac{f(x-q_n)}{2^n}$. Question: ...
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28 views

limit points in a topological space

I'm taking a topology course with the textbook by Munkres. And I saw the definition of convergence of a sequence and limit point of a subset of a given topological space. I have told in the class ...
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41 views

Support function of a set: when we can replace the supremum with the maximum?

Consider a set $K\subseteq \mathbb{R}^d$ (not necessarily closed and/or convex). Let $k:=(k_1,...,k_i,...,k_d)$ be a generic element of $K$. Consider the function $h_K:\mathbb{R}^d\rightarrow ...
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23 views

On the completeness of Weak Operator Topology

Let E,F be any two Banach space and let $\mathcal{B}(E,F)$ be the space of all bounded linear operators from E to F. I can show that this space is a complete space with respect to the norm and strong ...
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29 views

Showing $D = \{ f : |f(x) - f(y)| \leq \sqrt{|x-y|} \}$ where $D \subseteq C[0,1]$ is not bounded

Given $D = \{f: |f(x) - f(y)| \leq \sqrt{|x-y|}\}$ where $D \subseteq C[0,1]$, I am asked to show this set is not compact. To do so, I must show the set is not bounded (specifically not uniformly ...
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23 views

Regular closed sets in a subspace of a topological space

Suppose $\langle X,\mathscr{O}\rangle$ is a locally compact and Hausdorff topological space and $\operatorname{RC}(X)$ is its algebra of regular closed sets (i.e. these that are equal to the closure ...
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32 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
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36 views

Show that minimum and maximum are contained in $V(\mathcal{R})$/ Stones' axiom

Let $\mathcal{R}\subset\mathcal{P}(\Omega)$ be a ring for some set $\Omega$. Consider $$ V:=V(\mathcal{R}):=\left\{\sum_{i=1}^n\alpha_i1_{A_i}: \alpha_i\in\mathbb{R}, A_i\in\mathcal{R}, ...
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22 views

Getting topological objects from the “cube” of $T^3$

One can imagine $T^3$ much like he can imagine $T^2$: as a flat box with opposite faces identified. One may put coordinates on $T^3$, each of which would logically range from $0$ to $2\pi$. To get ...
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11 views

Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times ...
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22 views

Help showing compactness of the support of a function in the Sobolev Space $W^{1,p}$

In Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis, in the proof of Theorem 8.12, it is needed to show that the support of a function is compact. The function ...
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24 views

Closed subgroup implies open?

If $H$ is a closed subgroup of a topological group, $H$ is also open?, I know that an open subgroup of a topological group is also closed, but the converse is true? if isn't, wich could be a ...