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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Is there a theorem relates continuity of x-section&y-section of a function and continuity(measurability) of a function itself?

Let $(X,\tau),(Y,T),(Z,O)$be topological spaces. Let $f:X\times Y\rightarrow Z$ be a function. Let $f_x,f^y$ denote $x,y$-section of $f$ respectively. Let's assume $f_x,f^y$ are continuous for all ...
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19 views

On the definition of (small) inductive dimension

A regular topological space $X$ has inductive dimension smaller or equal to n if and only if: ($n=-1$) $X=\emptyset$; ($n>-1$) The space $X$ has a base of opens $\mathscr{B}$ such that, for all ...
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64 views

is $t: \frac{U}{U\cap V}\to \frac{UV}{V}$ not a homeomorphism of topological groups in gerneral?

If U and V are topological subgroups of a topological group G, such that G=UV, and V normal in G, we know that the map $t: \frac{U}{U\cap V}\to \frac{UV}{V}, x(U\cap V)\mapsto xV$ is a isomorphism of ...
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53 views

understanding of separable space

i would like to understand correctly what does mean separable space,from wikipedia i am reading that In mathematics a topological space is called separable if it contains a countable, dense subset; ...
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44 views

Prove that A has a geometric realization in $\mathbb{R}^d.$

A flag in a simplical complex K in $\mathbb{R}^d$ is a nested sequence of proper faces, $\sigma_0 < \sigma_1 < ... < \sigma_k$. The collection of flags forms an abstract simplical complex A ...
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76 views

Showing $int(A \cup B) \supseteq int(A) \cup int(B)$

Showing $Int(A \cup B) \supseteq Int(A) \cup Int(B)$ As $A \subseteq A \cup B$ we have that $Int(A) \subseteq A \subseteq A \cup B$ Similarly, $Int(B) \subseteq A \cup B$ $\implies Int(A) \cup ...
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23 views

Proving that products of paths respects homotopy rel endpoints

I'd like to show that given paths $f \cong g$ and $f' \cong g'$, homotopic relative their endpoints, then $f \cdot f' \cong g \cdot g'$ assuming of course that the $f$ ends where $f'$ begins, and ...
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36 views

Completeness and being totally bounded

Are completeness and being totally bounded somehow related with each other for uniform spaces?
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16 views

Set of $x$ such that $h \mapsto hx$ is proper

Let $X$ be a locally compact second countable space, and $G$ a locally compact second countable group wich operates continuously on $X$. If $x \in X$, let $\rho_x : g \mapsto gx$. I would like to know ...
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19 views

When is a locally metrizable space actually metrizable?

A topological space is metrizable when there is a metric that induces the topology. A topological space is locally metrizable when every point has some neighbourhood that is metrizable. According to ...
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487 views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
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19 views

Continuity of covariance kernels

Let $I$ be a locally compact Hausdorff (LCH) topological space. Let $c : I \times I \to \mathbb R$ be a covariance kernel, that is, a symmetric, nonnegative-definite function. Does it follow that $c$ ...
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47 views

Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff.

Let $R$ be a conmutavive ring. Prove the following: If $Spec(R)$ is $T_1$ then $Spec(R)$ is Hausdorff. Here $Spec(R)$ means the Zariski topology over the set of all prime ideals of $R$. To put it ...
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71 views

Complete unique geodesic space but not proper

Please comment on the example below about a complete unique geodesic space but which is not proper : Let $Y_n$ be the subspace of $S^2$ obtained by taking a sector bounded by two arcs of great ...
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57 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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49 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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21 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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50 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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35 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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31 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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81 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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30 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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57 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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40 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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47 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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40 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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107 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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124 views

Complement of set of all condensation point for an uncountable set of reals is at most countable.

Perfect Set: A set $E \subset X$ is said to be perfect if $E$ is closed in the metric space $(X,d)$ and every point of $E$ is a limit point of $E$. Condensation Point : A point $p \in X$ is said to ...
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54 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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33 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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34 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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25 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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64 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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59 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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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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53 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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29 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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105 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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56 views

Construct a topological manifold which its open cover is locally finite but not globally

The whole question is like this: 1-4. Let M be a topological manifold, and let U be an open cover of M . (a) Assuming that each set in U intersects only finitely many others, show that U is locally ...
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25 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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30 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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62 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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77 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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70 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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61 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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38 views

The characterization of asymptotic dimension

Let X be a metric space. The following conditions are equivalent (a)asdimX = n (b)n is the smallest integer such that for every R > 0 there exists n + 1 families Ui i=0,1,2,...,n, and S > 0 such ...
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40 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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55 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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64 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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82 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 ...