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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On characterization of Riesz homomorphisms on $C(X)$ space

How to prove the following: Let $K$ be an arbitrary topological space and $\pi: C(K)\to\mathbb R$ be a map with $\pi (1) = 1$. If $\pi$ is a algebra homomorphism then it is an Riesz homomorphism.
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28 views

Canonical choice of inverse system for profinite set.

Let $X$ be a profinite set - an inverse limit $\varprojlim X_i$. How can one prove that then $X=\varprojlim Y_i$, where $Y_i$ is finite quotient spaces of $X$? I may prove it if $X$ is topological ...
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41 views

Which part of differential geomety uses metrization theorems?

I learned three metrization theorems last year, which are Nagata-Smirnov,Smirnov and Bing. I thought these theorems are purely topological theorems, but i recently saw a post which says these ...
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123 views

Show that an hyperplane is closed iff f is linear and continuous

I need an help with the following exercise. Let $(E,\| \cdot \|)$ a n.v.s. and let $f:E\rightarrow \Bbb R$. Show that $H=\{x\in E: f(x)=\alpha\}$ is closed if and only if $f\in E'.$ Actually, I ...
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35 views

A space S is connected, if except for 0 and S itself, it has no subset whose boundary is empty?

The sphere S2 embedded in E3 is its own interior and closure and hence it has no boundary. An embedded S2 is a subset of E3 with empty boundary. This seems to imply E3 is not connected. What am I ...
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53 views

Classifying topological spaces by measures

While looking at some spaces, I happend to know, that in some spaces (like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ ...
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55 views

When is an outer Borel regular measure (inner and outer) regular?

Let $X$ be a topological space and $\mu$ an "outer" Borel regular measure on $X$ (for all $A\subset X$, there is $B$ Borel with $\mu(A)=\mu(B)$). Assume that $X=\cup _{i=1}^\infty U_i$, where each ...
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45 views

Radial Division of a Figure into Equal Parts

Given an $n$-gon, $P$, for which numbers $k$ must there exist a point $x$ so that there are $k$ equally spaced rays emanating from $x$ which divide $P$ into $k$ equal area parts? For $k=2$ we can ...
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81 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
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41 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
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61 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
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46 views

A topology defined on collections of open covers of a topological $X$.

Is anyone familiar with a topology which is defined on collections of open covers of a topological space $(X,O)$? I am trying to define a topology induced by a linear ordering of the open covers, ...
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77 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
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109 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
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23 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
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31 views

Completeness of Locally Compact Metric Space and Group of Isometries

Let $X$ be a locally compact metric space, and suppose that the group of isometries of $X$ acts transitively. Show that $X$ is complete. (This is 2nd part of a problem. In first part I showed that for ...
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26 views

Mapping open on open dense subset => Mapping is open on whole space?

Let $X,Y$ be topological spaces, and let $f\colon X \to Y$ be a continuous function. Further suppose that there exist an open and dense subset $U$ of $X$, such that $f\vert_{U} \colon U \to Y$ is an ...
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83 views

Closed and Connected subgroups of $\mathbb{R}^n$

Question is : What are closed connected subgroups of $\mathbb{R}$ and from that deduce what are closed connected subgroups of $\mathbb{R}^n$ What i have done so far is : Only connected subsets of ...
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31 views

Fundamental group of D^2

Could someone please help me to calculate the fundamental group $\pi_1\left(\mathbb{D^2}/(x\sim ix \text{ for } x\in S^1),0\right)$ without using Seifert? Maybe via some deck transformation group?
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88 views

Has anyone seen this space before? Does it have a name?

See the space below (the set taken as a subspace of the plane). It sort of looks like a comb, but with the wrap-around portion added, and the lower left corner removed. What would be a good name ...
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45 views

Extending a homeomorphism between subspaces

Lavrentiev's Theorem. Suppose $X$ and $Y$ are complete metric spaces, $A\subseteq X$, $B\subseteq Y$, and $f:A\to B$ a homeomorphism. Then $f$ can be extended to a homeomorphism $\overline f :G\to ...
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39 views

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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50 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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47 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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37 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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56 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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27 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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39 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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44 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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43 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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162 views

Nerve Theorem: Is the finite union of closed convex sets triangulable?

My Question: Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable$^1$? If so, why? Background: I'm trying to better understand the ...
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36 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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63 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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33 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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57 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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17 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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33 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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24 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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37 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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32 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 $\tilde{F}: [0,1]\times X\to ...
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$\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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67 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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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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41 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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66 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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105 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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25 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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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 ...