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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68 views

Characterizing $C_p(X)$ for regular or Tychonoff space $X$.

Are there any known characterizations for $C_p(X)$ Given that $X$ is regular or even Tychonoff? $C_p(X)$ is the space of continuous real valued functions $f:X \rightarrow \mathbb R$ with the topology ...
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63 views

Weak Compact and separable sets

Is true the following statement? Se $(X\|\cdot\|)$ a Banach espace, and $K\subset X$ a convex, weakly compact and separable set. Let $x_{n}$ a sequence in $K$. Thus, given any $\epsilon >0$ there ...
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24 views

Concordant maps and their $n$-th powers

Let $f:X\to Z$ and $g:Y\to Z$ continuous maps where $X$ is a subspace of $Y$ and $f=g\restriction _X$. Then $f$ and $g$ are called concordant if $f^{-1}(z)$ is a dense subset of $g^{-1}(z)$ for every ...
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58 views

$\sigma$-$\sigma$-compactness is $\sigma$-compactness?

I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact? I'm not sure if a countable union of countable unions is still a countable ...
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46 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
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35 views

disjoint compact subsets of $\ell^\infty(\mathbb{R})$

What would be some compact subsets of $\ell^\infty(\mathbb{R})$ which are disjoint? I know that the set of convergent sequences in $\mathbb{R}^n$ is one compact subset, but what would be another which ...
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88 views

Connectedness in proximity spaces

Let $\delta$ be a proximity. A set $A$ is connected regarding $\delta$ iff $\forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right)$. ...
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33 views

Show that un in a compact topological space, any infinite set has some limit point. When does the reverse hold?

Show that un in a compact topological space, any infinite set has some limit point. When does the reverse hold? My attemp: I have done the proof but i dont know when the reverse hold. I have come up ...
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75 views

Resolvable spaces

a space $X$ is called a resolvable space if it is expressible as a union of two disjoint dense subsets. I want to find a resolvable but not lindelof space? Is there any example such a space?
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49 views

Sets that have the property of Baire

Can I say that a set $A$ has the property of Baire, if and only if it is of the form $A=(B \setminus C) \cup D$ where $B$ is regular open and $C,D$ are of first Category? Are there any other useful ...
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51 views

What are the compact sets in chains or posets with either the left or interval topology?

In the real line in the usual topology the compact sets are the closed and bounded sets. In the left topology, the topology generated by the left-open intervals $(-,a)$, the compact sets are exactly ...
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56 views

A “complementary” topology

If $(X, \tau)$ is an Alexandrov topology then arbitrary intersection of open sets are open, and likewise arbibtrary unions of closed sets are closed, so we can define a topology $(X, \tau')$ as $$ U ...
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50 views

G-space decompositions preserved by equivariant maps?

Let $X,Y$ be topological $G$-spaces, with (left) $G$-invariant probability measures $\mu_X,\mu_Y$ respectively, and let $f:X \to Y$ be a surjective $G$-equivariant map preserving the measures, i.e. ...
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210 views

Brouwer's fixed point theorem on a sphere

Let $f: S^{2} \rightarrow S^{2} $ be a continuous map such that there exists a closed "disk" $D$ that is mapped to itself. Then will $f$ have a fixed point?
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66 views

A criterion for compact topological spaces

A book I'm reading uses the following argument, but I could't see how it works: Suppose $X$ is a topological space, $\mathcal{A}$ is a pre-basis for the topology (i.e. the topology on $X$ is the ...
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83 views

$\mathcal{N}^\omega$ is homeomorphic to $\mathcal{N}$, where $\mathcal{N}$ is Baire space.

I am trying to prove that $\mathcal{N}^\omega$ is homeomorphic to $\mathcal{N}$ where $\mathcal{N}$ is Baire Space $\omega^\omega$, of all sequences of natural numbers, $\langle a_n;n \in ...
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37 views

Cylinder as Fibre bundles

I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can ...
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55 views

When do continuous surjections have Borel sections?

It is known that whenever we have a continuous, surjective map $f\colon X\to Y$ between compact metrisable spaces, there is a Borel (even Baire class $1$) section $g\colon Y\to X$ (so that $f\circ ...
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28 views

Topology - Connected Images

Let X be a topological space and let Y = {0,1,2} have the D topology. Assume f: X$\rightarrow$Y is a continuous function. If A is a connected subset of X, what are the possible values of the image ...
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71 views

Abelianized fundamental group

Let $P$ be the projective plane and let $nP$ be the connected sum of $n$ copies of the projective plane. Show that the abelianized fundamental group $\pi_{1}(nP)/[\pi_1,\pi_1]$ is the direct sum of a ...
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72 views

Mixed dimension non-Euclidean geometry?

Is the following a "consistent non-Euclidean geometry"? It seems to satisfy the first 4 Euclidean postulates. Any comments? Any agreements or disagreements? Following are the additional conditions on ...
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100 views

Hausdorff and Quotient Spaces

Let $L$ be a subset of $\mathbb{R}^{2}$ and let $N = \mathbb{R}^{2}/L$ be the quotient space obtained by identifying all points in $L$ to a single point. I need to prove that $N$ is Hausdorff ...
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56 views

Topology generated by a family of maps and a similar question for measure theory

Let us say that $X$ is a set, $f$ from $X$ to some topological space $Y$, and we endow it with the smallest topology for which $f$ is continuous. Is it true that for any $f_1:X \rightarrow V$ with ...
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80 views

Maps $S^1 \to S^1$ of equal degree are homotopic.

Let $a(t)=e^{2\pi it}$ be the generator of $\pi_1(S^1,1)$ and we define degree of $f$ this way: $$[\omega^{-1}]\cdot f_*([a])\cdot [\omega]=\deg(f)[a]$$ where $\omega$ is any path from $f(a)$ to ...
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25 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
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82 views

$C(X,Y)$ complete

I want to prove that: $C(X,Y)$ is complete in the compact-open topology, when every component of $X$ is locally compact with a countable base, and $Y$ is a complete metric space. The proof I am ...
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34 views

Existence of slices for the action of a subgroup

Assume that a group $G$ acts on a space $M$ in such a way that there exists a slice at a point $m \in M$. Let $H \subseteq G$ be a subgroup. Under which additional assumptions (if there are any) can ...
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207 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
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27 views

Self similar set and its measure

Prove: If $(K,\{f_i\}_{i=1}^N)$ is a self-similar set and $(\mu,\{\mu_i\}_{i=1}^N)$ is a self-similar measures, there is any arbitrary partition $\Lambda=\Lambda_a(r_1,\cdots,r_N)$ and ...
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34 views

Convergence of a Sequence of Functions

The context: verifying the group axioms for the fundamental group, specifically that every element has a unique inverse. Below is a non-example, and I am tasked with explaining $why$ it fails. Let ...
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69 views

What is the picture for $S^1\times S^2$?

Let M be a compact, connected, two-dimensional manifold such that $M = S^1\times S^2$. How should one picture M? The following is from Abraham and Marsden, Foundations of Mechanics: Let $M$ be ...
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96 views

Relative Interior and dense subsets

(Due to no answers, I also posted this question here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ ...
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116 views

Characteristic properties for topological pushouts and pullbacks

So far in my topology class we've talked about several topological constructions (namely the subspace topology, the quotient topology, the (finite and infinite) product topology, and the disjoint ...
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78 views

Visualizing a Curve Drawn on Plane Model of Sphere

The following is an image from Sue Goodman's $Beginning$ $Topology$. My question: if I were to draw this on the space model of the sphere, am I correct that it would section the sphere into ...
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43 views

What is the name of $C(A)/A$

Given a topological space $A$, $C(A)$ is the cone of $A$. The space $C(A)/A$ is clearly homotopic to the suspension. My question is if it has a widely known name?
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115 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
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88 views

compactness and boundeness

Use the open cover characterization of compactness to prove that if $f : [a,b] \to X$ is a continuous function and X is a metric space, the f is bounded. Ok so I have a different approach. Does this ...
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36 views

If $N$ is path-connected, and $N^C$ is path-connected, a neighborhood of $N$, $M$ has $M-N$ as path-connected

So I have been doing some topology and I came up with this question, and can't find a simple way to prove it. I can kinda make the argument that if we have two points in $M-N$, the are connected by a ...
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36 views

Another question on spaces with calibre-$\aleph_1$

Let $X$ be a strongly monotonically monolithic space with calibre-$\aleph_1$. Must $X$ be Lindelof? I know $e(X)=l(X)$ for a strongly monotonically monolithic space. So to prove that $X$ is ...
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72 views

Find the orbit space $T^2 / \mathbb Z_2$

Let $T^2$ be the unit torus $$ T^2 = \left\{ (\lambda, \lambda') \in \mathbb C^2 \mid |\lambda| = |\lambda'| = 1 \right\}. $$ Then the group $\mathbb Z_2$ is acting on $T^2$ by the rule ...
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113 views

Proof that a set $X \subset M$ is a Manifold

Let M be a manifold without boundary and let , $g:M\to \mathbb R$ have $0$ as a regular value. Than the set $X \subset M$ with $g(x) \ge 0$ is a smooth manifold with boundary equal to $g^{-1}(0)$. I ...
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46 views

Graphs, Tiling and topological equivalence

Let $X \subseteq \mathbb{R}^2.$ I am fully aware that no subset of the plane it topologically equivalent to a torus. However, it seems to require a lot of heavy machinery to prove this. Does it ...
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107 views

Alternative proof to Urysohn's lemma using $d(x,A)$.

Is there an alternative proof to Urysohn's lemma, that makes use of $d(x, A)$? Urysohn's lemma is: given a normal topological space $X$, for any disjoint closed sets $A$, $B$, there exists a ...
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76 views

Topology of Convergence

I am having some difficulties in understanding the concept of topology induced by convergence? especially how the weak convergence induces weak topology? Does anybody know a good reference which ...
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84 views

About definition of topology

let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if: 2)$ \forall x \in X (\forall t \in m_x(x \in t)) $ 3)$ ...
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154 views

Proofs that quasicomponents of compact Hausdorff spaces are connected

Nuno's answer to Any two points in a Stone space can be disconnected by clopen sets uses (and proves) the following: Theorem: Let $X$ be a compact Hausdorff space. Then the quasicomponents of $X$ are ...
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48 views

A topological space is called $T_B$ if…

A topological space is called $T_B$ if every compact subset is closed. According to therem ( I, II , III), how does the below theorem proof?? Let $(X,\tau)$ be a $T_B$-space which is not ...
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310 views

Subsets of $\mathbb{R}^2$ that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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92 views

Connected sum of two surfaces is a surface?

IS the connected sum of two surfaces a surface? Im having hard time trying to see this. Can someone kindly help me? thanks.
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95 views

Let $\mathcal{F}$ be a filter on $X$. $\mathcal{B} \subseteq P( X)$ is called a filter- base…

Let $\mathcal{F}$ be a filter on $X$. $\mathcal{B} \subseteq P( X)$ is called a filter- base satisfies bellow conditions: ( 1 ) : ‎$ ‎\mathcal{B}‎ ‎\neq‎ ‎\emptyset‎ $‏ ‎‎( 2 ) : ‎$ ‎\emptyset ...