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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Space of measures is weak-* Hausdorff?

If $X$ is a topological space which is hereditarily Lindelöf and completely regular, then the space of finite signed measures on the Borel $\sigma$-algebra, endowed with the weak-* topology, is ...
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51 views

Prove: If H and G/H are totally disconnected then G is also totally disconnected

Let $G$ be a topological group and $H$ a subgroup of $G$. If $H$ and $G/H$ are totally disconnected, then $G$ is also totally disconnected. With 'totally disconnected' we mean the every connected ...
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41 views

Covering of Riemann sphere

The question consists of several parts: What is the simply connected ramified covering of the Riemann sphere with ramification indexes {2. 3. 5} over three points of RS in every preimage of these ...
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48 views

Any finite-dimensional subspace of a Hilbert space is closed: easier proof?

A noted theorem is that a finite-dimensional subspace of a Hilbert space must be topologically closed. I have seen some proofs of this theorem which are less simple than this, but what is wrong with ...
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38 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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57 views

Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?

Moderator's note: I've copied the title of the post from a more recently posted version of this question by the OP. The text below is from the original question: I am not sure whether the ...
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34 views

Principal order filters on a POSET

I have another problem, but in this one I have no idea how to start. Let be $(X,\leq)$ a POSET with a first element and gifted with the topology $\{ (a,\rightarrow) : a \in X \}$ (principal order ...
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43 views

every path in $X$ is homotopic with endpoints fixed to a path passing through $b$

$X$ is path connected and b$\in$X, show every path in $X$ is homotopic with endpoints fixed to a path passing through $b$ This is the hint in the book: Let $\gamma$ be a path from $x$ to $y$. If ...
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45 views

Prove that the density topology is stronger than Euclidean topology.

I'm working through Franklin Tall's paper on the Density Topology. In theorem 2.3, he defines a topology on $\mathbb{R}$ such that if a set $E \subseteq \phi(E)$, then $E$ is open. He uses $\phi(E)$ ...
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44 views

Show that the following Statements is true?

Let $\tau $ be the topology on $\mathbb R$ for which the interval $[a,b)$ form a base.Let $\sigma$ be a topplogy on $\mathbb R $ such that $\tau \subseteq \sigma$. Then If the map $ x \mapsto -x$ ...
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44 views

Topological entropy, spanning sets and expansiveness of simple maps on a torus

I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$. By induction, we have ...
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44 views

Some basic Topological proof help please.

Basically im really bad at proofs and i havent done math in almost a year and decided id like to learn topology on my own... just want someone to be really critical on my solutions please also i would ...
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67 views

“forgetting base point”-map properties

I need help with the following problem: Let $P:\pi_1(X,x_0)\to S(X)$ be the map from the set of homotopy classes of loops based at $x_0$ to the set of homotopy classes without restriction on the ...
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28 views

Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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37 views

Is there a way to check if 2D level set function has changed from representing an object of genus 0 to genus 1?

This question has been moved from stack overflow to here. My goal is image segmentation but I think my question is a math one: In computer vision level sets are regularly used to represent moving ...
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68 views

Open ball in infinite dimensional Banach space is not weakly open

I have to prove that open ball in infinite dimensional Banach space is not weakly open. I have no idea how can I do it. I think that I should reach contradiction with infinite dimensions.
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19 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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37 views

Grassmannian Manifold homeomorphism

I have troubles understanding the meaning of this excercise: I am supposed to show that the Grassmannian manifold $G_{k,n}$ of k-dimensional subspace in $\mathbb{R}^n$ is homeomorphic to $O(n)/(O(k) ...
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25 views

symmetric quasi-uniformity

A quasi-uniformity $U$ will be called symmetric provided that $U = U^{-1}$, that is, provided that it is a uniformity. Otherwise it will be called nonsymmetric. It is readily seen that the supremum ...
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40 views

Continuity of a function in the product topoogy

Hi everyone I would like to understand if my reasoning is correct. Let $X$ be the space of sequences with values in the interval $[0,1]$, i.e. if $\mathbb{N}$ is the set of natural numbers, $x\in X$ ...
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35 views

Can a factor map be a Serre fibration?

Let $D_n$ be an $n$-disc. Is the factor map $p: D_n\to D_n/S^{n-1}\simeq S^n$ a Serre fibration, in other words, can any homotopy $F: [0,1]\times X\to S^n$ be lifted to $\tilde{F}: [0,1]\times X\to ...
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22 views

$\omega$-covers and $S_1(\Omega,\Gamma)$ property

I am readund this article and there is some proof in there (top of page 156) which is not clear to me. The definitions are: 1. Property ($\gamma$): If $\mathcal U$ is an $\omega$-cover of $X$, ...
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73 views

Sets which are open “modulo a nullset”

A set $A$ is said to have property of Baire there exists an open set $U$ such that $A\triangle U$ is meager. So this says that symmetric difference of $A$ and some open set is small (in the sense of ...
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44 views

Let $X$ a finite set, and $X^{*}=X\cup \{\omega\}$ wiht $\omega\notin X$. Given a filter $\mathcal{F}$ on $X$

Let $X$ a finite set, and $X^{*}=X\cup \{\omega\}$ wiht $\omega\notin X$. Given a filter $\mathcal{F}$ on $X$, Show that $$\mathcal{T}(\mathcal{F}):=2^{X}\cup\{F\cup \omega\mid F\in\mathcal{F}\}$$ ...
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45 views

If a basis is also a subbasis for the same topology, is it closed under finite intersections?

I'm trying to understand a little bit the relation between a basis and a subbasis for a given topology. So suppose $S$ is a subbasis for a topology, say, $\delta_s$. Suppose that $S$ is also closed ...
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69 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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120 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
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26 views

The game $G(K,X)$

In Telgarsky - Topological games, in page 246, the following game $G(K,X)$ is described: There are given a space $X$ and a class $K$ of spaces such that $Y \in K \Rightarrow \mathcal F(Y) \subset K$. ...
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25 views

Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
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61 views

Locally compact space .

How can I show that for every locally compact space $X$ there exists a one-to-one continuous mapping of $X$ onto a compact space. And is it necessary that $X$ is compact since we know that every ...
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48 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
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31 views

Neighborhoods for continuous functions between CG spaces

I have a couple of problems regarding the existence of certain neighborhoods, so as to prove continuity of suitable functions. Suppose then that $Y,X$ and $Z$ are compactly-generated Hausdorff spaces ...
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31 views

Does any $\omega$-cover in which $X \in L(\mathcal U)$ is also a $\gamma$-cover?

As a continuation to this question: An $\omega$-cover, is an open cover $\mathcal U$ of $X$, such that, $X \notin \mathcal U$, and for every finite set $F \subset X$, there exists an open set $U ...
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62 views

Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
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40 views

$\overline{A}=A\cup \delta A$ proof

Can somebody help me out with proving the following equality? $\overline{A}=A\cup \delta A$ where $\delta A=\overline{A} \cap \overline{A^c}$.
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37 views

Topology - Compactness of $\mathbb{Z}\times\{0,1\}$

A question from my h.w.: Is the topological space $\mathbb{Z}\times\{0,1\}$ (where $\mathbb{Z}$ has the discrete topology and $\{0,1\}$ the trivial one) compact? sequentially compact? ...
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86 views

Hanging a picture with Beta functions

There's a classic puzzle that goes something like this: You have two nails in a wall, and you want to hang a picture with a string (think of a necklace with a pendant) in such a way that if you ...
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46 views

What is wrong with this proof in topology?

Let $X$ be a $T_4$-space and $M \subset X$, then $M$ is a subspace that is also $T_4$. Proof: If $A,B$ are closed in $M$, then $A=W_A \cap M$ and $B = W_B \cap M$ for some closed sets $W_A,W_B$. ...
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Compactification via embeddings and extending continuous functions

My question comes from reading Munkres' Topology, the section on Stone-Čech compactification. To find the compactification $\mathrm{Y}$ of $\mathrm{X}$, we find an embedding h, $\mathrm{h}: X ...
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28 views

Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
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44 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
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87 views

Nagata Smirnov Metrization Theorem

I am looking for a proof for Nagata-Smirnov Metrization Theorem, but I couldn't find one that is readable. I found the paper by Nagata written in 1954 but it is unreadable and uses old notation. ...
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50 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
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29 views

Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
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Neccessary and sufficient conditions to form a topological ring on $\Bbb{Z}$?

Let $B = \{ \{a + b f_i(n) : n\in \Bbb{Z}\} : a,(b\neq 0) \in \Bbb{Z}, f_i \in F \}$. Then what are necessary and sufficient conditions on the set of integer functions $F$ such that $B$ is a basis ...
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28 views

All topology pairs $(X,Y)$ such that $f: X \to Y$ is continuous.

Given an arbitrary function, or more specifically if you want let $R$ be a ring and let $X = S \times S; Y = R; S \subset R$ and $f(a,b) = a - b$, is there something interesting about all the topology ...
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Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
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32 views

How to show that these two constructions of Tychonoff product topology are equivalent?

Definition: The Tychonoff product topology on $X = \Pi_{t \in T}X_t$, is the topology $\tau$, which is generated by the family $\bigcup\{ p_t^{-1}(\tau(X_t)) : t \in T \}$ as a subbase where ...
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104 views

n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
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28 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...