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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94 views

Properties of induced Hausdorff metric space.

M is a metric space. Let K be the set of closed, compact and nonempty subsets of M. Define Hausdorff metric on K. Then the topology on K is the topology induced by Hausdorff metric. Prove that if M ...
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23 views

Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...
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173 views

how to evaluate these definite integrals involving the following hypergeometric function

Could you please help me with this integral? I want to evaluate the following integral $$ I_0=\int_{-\frac{1}{2}arccosh(\frac{1}{2 \alpha^2})}^{\frac{1}{2}arccosh(\frac{1}{2 \alpha^2})}(2\alpha^2 ...
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87 views

Action of the fundamental group on a Universal cover

Let $\pi: \tilde{X} \mapsto X$ an universal cover. I know that $\tilde{X}/Aut(\tilde{X},\pi) \simeq X$. Let $H \subset \pi_1(X,q)$ a subgroup of the fundamental group and consider the orbit space ...
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63 views

When proving that the translate of a closed set is closed in $\mathbb{R}$, do I need to consider the case when the translate is infinte?

Let's say I have a closed set $A \subseteq \mathbb{R}$. If I take the translate of $A$ with respect to some real number $y$, will I have to consider the case when $y = \infty$? I do not think I do ...
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44 views

How do i directly prove Urysohn's Lemmas from one to another?

It's possible to prove "Urysohn's lemma for locally compact Hausdorff" from "Urysohn's lemma for Normal space". But how do i prove the converse? I want to know which one is more general.. Thank you ...
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28 views

Is it possible to prove product of basis is a basis for the box topology without AC?

Let $\{(X_i,\tau_i)\}$ be a collection of topological spaces and $\mathscr{B}_i$ be a basis for $\tau_i$. Let $\mathbb{B}=\{\prod p_i : p\in \prod \mathscr{B}_i\}$. Then is $\mathbb{B}$ a basis for ...
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38 views

I want to find a topologicaly embedding $f : X \rightarrow Y$ and $g: Y \rightarrow X$, yet $X$ is not homeomorphic to $Y$.

This is related to a problem of showing that none of the intervals $(0,1], (0,1), [0,1]$ are homeomorphic to another, since it's in the same problem block. I've tried using $f(x) = 1 - x \ , f(x) = ...
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73 views

Guess that Topology

I am interested in a geometry that I'm imagining but don't possess the requisite language to understand completely. I'm searching for an answer which points me in the right direction. The space I'm ...
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18 views

Chromatic classes of vertices of a polyhedron

For a convex polyhedron, how do I figure out all possible proper chromatic classes of its vertices (so that all vertices that are assigned the same color constitute a separate class, and no two ...
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76 views

topological space $T_B$

a topological space $X$ is called $T_B$ if every compact subset is closed. Is $X \times X$, $T_B$ when $X$ is $T_B$ ? If it is not true, which conditions do need to $X \times X$ be $T_B$?
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57 views

Usual topology of the set…

What is the usual topology of the set formed by the union of line segments in $\mathbb{R^2} $, which join the points of $ \mathbb {Q} $ on $ [0,1] $ to the point $ p = (0.1) $.
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45 views

Proving that all delta-nbhd are open sets by using the contrapositive of the definition of an open set

So I want to prove that all $\delta -nbhd$ are open sets the following way: Let$\space E$ a $\delta -nbhd$ Take a point $m$ that is not an interior point of $E$ and prove that $m \notin E$ Here's ...
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55 views

An exercise on Lebesgue measure.

I'm studying for my exam, and I found this problem: Let $A$ be a borealian set of extended $\mathbb{R}$ . Show that $A$ has Lebesgue measure $0$ iff for all $\epsilon>0$, exist disjoint open ...
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42 views

Fixed Point Property in $\Bbb{R}$ and $\Bbb{S}^1$

Do $\mathbb{R}$ and $\mathbb{S}^1$ enjoy the fixed point property? I am trying to find a counter-example, but I can't think of one now. Probably they both have the fixed point property. But how do we ...
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19 views

isolated zero z of X on a star shaped polygon

Consider a polygon that is star shaped with respect to the isolated zero $z$ of $X$. I want to show that the boundary of the polygon can be made transverse to $X$ by jiggling vertices only in the ...
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129 views

If $B$ is sequentially compact and $A \subseteq B$ is closed, then $A$ is sequentially compact

$DEF:$ A set $X$ is sequentially compact if $\{x_k\}_{k \geq 1} $ is a sequence and $x_k \in A$ for each $k$, then there is a point $x \in A$ so that $x$ is a limit point of $\{x_k\}_{k \geq 1} $ ...
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129 views

Regarding this proof that the union of bounded sets is bounded (generalized metric space)

http://www.proofwiki.org/wiki/Finite_Union_of_Bounded_Subsets I am having trouble seeing how this proves the idea at all. It says: $$d(x, a_1) \le d(x, a_2) + d(a_1, a_2)$$ so $$d(x, a_1) \le K_2 + ...
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46 views

Showing something is homeomorphic to $S^2$.

Suppose $X,Y$ are compact surface such that $X\#Y \approx X$ for any compact $X$. Show that $Y$ is topologically equivalent to the sphere. I was thinking for a while about this. It seems pretty ...
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52 views

multi has closed graph

Please help me prove this Let $\Omega\neq\emptyset$ be a subset of a Banach space, X be a Banach space and $F:\Omega\to 2^X\setminus\emptyset$ have closed values. Then (a) If $F$ is ...
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18 views

Posets and Level Sets, Pt 2

We consider the cube $L=[0,r_0]^3$ where $r_0 > 0$ is a fixed real number. The lattice is generated by the level set $L_r$ We define the level function $\lambda (x,y,z) \to x+y+z =r$ For $ r ...
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196 views

Closed subspace of $C([0,1])$.

Consider the space $C([0,1])$ of continuous real-valued functions on $[0,1]$ with metric $d(f,g) = \max_{x \in [0,1]}|f(x) - g(x)|$. Let $A = \{x^n : n \in \mathbb{N}\}$. I want to prove that $A$ is ...
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57 views

If $Z$ is a retract of $Y$ and $Z$ is contractible then $Y$ is contractible.

In a proof it seems like they use this statement: If $Z$ is a retract of $Y$ and $Z$ is contractible then $Y$ is contractible. Is this true? I have tried to prove it without success and would ...
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21 views

$\mathbb{RP}^2$ and crosscap

Hmm, I wonder what is the relationship between $\mathbb{RP}^2$ and a cross-cap...? Is it like klein-bottle in $\mathbb{R}^4$, which they do not really intersect and embeds in $\mathbb{R}^3$, where ...
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39 views

Reeb Vector Field: Actual Construction in (R^3, Std) Contact Structure, given Open Book

say we have the Standard Contact Structure on $\mathbb R^3 , (r,\theta,z)$/~; $(r,\theta,z)$ ~ $(r,\theta,z+1)$, given by $ker(dz+r^2d\theta)$ ;we have an open book decomposition in which the pages ...
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65 views

Hartshorne II Prop 2.6

Prop 2.6 constructed a continuous map X to t(X), I cannot verify that it is a homeomorphism. I try to show any open set U is mapped to t(X)\t(X\U). To show it is surjective, let K be an irreducible ...
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83 views

Reference for closed map lemma

I would like to have a reference (book, page) for the following version of the closed map lemma: If a continuous function between locally compact Hausdorff spaces is proper (i.e. preimages of compact ...
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40 views

hereditarily Lindelöf , KC space

A space is said to have the finite derived set property if each infinite subset $A ⊂ X$ contains an infinite subset with only finitely many accumulation points in X. A hereditarily Lindelöf, ...
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105 views

Is the interior of simply closed curve homeomorphic to disk?

I am strongly guessing that the interior of any simple closed curve in $R^2$ should be homeomorphic to unit disc $D$. But I cannot prove it. Is it true? Then can you shed me some idea for proving ...
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36 views

A question on $G^*_\delta$-diagonal

Let $X$ be any toplogical space and $A$ be a closure of some discrete subset of $X$ with $|A|\le \omega_1$. If $A$ has a $G_\delta$-diagonal, does $A$ has a $G^*_\delta$-diagonal? A space $X$ has a ...
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43 views

Is a uniquely geodesic space contractible? II

Is a uniquely geodesic space, contractible ? With the extra assumption that closed metric balls are compact, there is an answer here. We expect here an answer beyond this extra assumption ...
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50 views

Linear map preserves closedness of a convex set with property on recession cone

Let $\mathbf{E},\mathbf{Y}$ be two euclidean space, $C$ be a non-empty closed convex set in $\mathbf{E}$. The map $A:\mathbf{E}\rightarrow\mathbf{Y}$ is linear, and $N(A)\cap 0^+(C)$ is a linear ...
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83 views

What's the classification of CW complexes formed by gluing a 2-cell to a circle?

After this answer, the following question comes : What's the classification (up to homeo.) of CW complexes formed by gluing a 2-cell to a circle ?
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75 views

Stratification of a smooth map

I am trying to do an exercise. Namely, find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+bx+cy$, where $a,b,c$ are parameters. As I have seen, this is also ...
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49 views

Hyperplane execises?

please what is the idesa to solve yhis exercise? why we must show the hint proposed ? Thank you
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74 views

Prove that function $g(x \times y) = x + y^{2}$ is a quotient map

I got this result from a problem in Munkres, but I can't prove it directly (by the definition). Hope some one here can help me. In space $R\times R$, consider a function $g(x \times y) = x + ...
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45 views

Clopen set in the Stone-Čech compatification

In the Stone-Čech compactification: What kind of structure do the clopen sets have ?
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53 views

countably compact , Ultrafilter

Let $(X,\tau )$ be a KC-space which is not countably compact. Then X can be condensed onto a weaker KC-topology. Proof: Let new topology $‎ ‎\tau‎^{‎\prime‎} = ‎\{U‎‎\in‎‎ ‎\tau:‎‎ ‎x‎_{0}\not\in ...
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100 views

Intersection of simply connected sets II

I read the following statement in the old question "Intersection of Simply-Connected Sets" (Intersection of Simply-Connected Sets): If $U$ and $V$ are simply connected and $U \cap V$ is path ...
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82 views

How to prove the contiunty of Thomae's by the concept of general topology?

The function is $f(x)= \begin{cases} 1/n \quad &\text{if $x= m/n$ in simplest form;} \\ 0 \quad &\text{if $x \in \mathbb{R}\setminus\mathbb{Q}$.} \end{cases} $ Show that $f$ is ...
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36 views

$E_1+E_2$ is open if both open?

if $X$ be a norm linear space and $E_1,E_2\subseteq X$ then $E_1+E_2=\{x+y:x\in E_1,y\in E_2\}$ is open if both open? is open if one is open and another is closed? closed if both are closed? I just ...
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55 views

first countability space

We know that $ [0,\omega_{1}‎)‎ $‎ is a first countable space. To show this we take $ ‎‎\mathfrak{B}(\alpha )‎ =‎ ‎\{ (\zeta,‎\alpha ] :‎ ‎‎\zeta <‎ ‎‎\alpha ‎ ‎\}‎ $ as local base. But I ...
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88 views

How to prove that a first-countable topological space satisfies “closed-compact” if and only if it is compact Hausdorff?

A topological space is called $C$-$C$ iff the closed sets in $X$ coincide with the compact sets in $X$. Let $(X,\tau)$ be a topological space which satisfies the first axiom of countability. Then ...
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53 views

Alternative definition of regular space.

A space where every singleton set is closed is called regular if for a pair of a point and a closed set we shall get two disjoint open sets one contains the point and the other contains the closed ...
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53 views

Spherical clipmap to heightmap mapping

I'm working on a library to create real time rendered planet models using spherical clipmaps. I can't seem to figure out how to map my heightmaps to the clipmaps and was hoping someone here could ...
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46 views

ultrafilter and compact

*Let ( X,τ ) be a KC non-compact space. Then there is a discrete subset $ D ⊆ X $, such that $ \overline{D } $ is not compact. Furthermore there is an ultrafilter F in X, such that $ D \in F $ and ...
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24 views

a question about support of a function on GL(n)

Let $f$ be a function on $GL(n,F)$, where $F$ is either a p-adic field, or $\mathbb{R}$. Let $N_n$ be the subgroup of upper triangular unipotent matrices. $\bar{N}_n$ be the lower triangular unipotent ...
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33 views

ultrafilter and KC minimalspace

By definition in a $KC$ space every compact set is closed. A space $(X,‎\tau)‎‎‎$ is $KC$-minimal if $(X,‎\tau )‎‎‎$ is a $KC$ space and there is no $KC$ space $(X,\sigma)$ such that ...
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39 views

Interval in Product Space

I am reading a paper now that refers to an "interval" in $\mathbb{R}^{[0,1]}$. But what does interval here refer to? Does this mean there is some ordering of the elements in $\mathbb{R}^{[0,1]}$? I am ...
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54 views

Is the following space $\sigma-$compact?

Let $X$ and $Y$ be two $T_{2}$ spaces, such that $Y$ is $\sigma-$compact. Let $f:X\to Y$ be an open continuous surjective map. Is $X$, $\sigma-$compact?