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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2answers
78 views

Show that a space is separable.

I want to show that the topological space $[0,1]^{[0,1]}$Separable space. That seems like a very complicated space...it is not countable, right? I have to find a set which is dense in the ...
2
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3answers
140 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
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2answers
84 views

If X is local compact, then it holds: A is closed $\iff$ $A\cap K$ is compact for all compact K [closed]

Prove: Show that for every local compact space X holds the following: A $\subseteq$ X is closed $\iff$ $A \cap K$ is compact, for all compact sets K. I use the following definition of local ...
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2answers
119 views

Why do we denote $S^1$ for the the unit circle and $S^2$ for unit sphere?

Maybe a quite easy question. Why is $S^1$ the unit circle and $S^2$ is the unit sphere? Also why is $S^1\times S^1$ a torus? It does not seem that they have anything in common, do they?
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1answer
153 views

Is $GL_2(\mathbb Z)\cdot X$ a dense subset of $\mathbb R^2$?

We know that the set $D=\{a+b\sqrt{2} \mid a,b\in \mathbb Z\}$ is dense in $\mathbb R$ because $D$ is a subgroup of $(\mathbb R,+)$ that is not of the form $\alpha \mathbb Z$. So, the following set ...
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0answers
46 views

The principle $S_1(\mathcal O,\mathcal O)$ versus the game $G_1(\mathcal O,\mathcal O)$

Given a topological space $X$, Let $\mathcal O$, denote the set of all open covers of $X$. We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ ...
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1answer
164 views

Irrational Rotation

Let $\sigma$ be a homeomorphism of $S^1$. Then the following statements are equivalent; (1) O(z) is dense in S for some z in S, (2) O(z) is dense in $S^1$ for every z in $S^1$, (3) $\sigma$ is ...
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1answer
714 views

best intuitive books/video lectures to read topology and functional analysis

What are the best intuitive books/video lectures to read topology and functional analysis ? I am aware of basic linear algebra, analysis and measure theory.
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1answer
60 views

Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
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1answer
46 views

Topology of Normed Space

$(X, \lVert \cdot \rVert)$ is a normed space. Let $x \in X \setminus \{0\}$ and $Y \subset X$ is a subspace. Prove that if $Y$ is open then $Y=X$. Which technique is more useful? We know ...
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168 views

Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that ...
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1answer
2k views

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a ...
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1answer
145 views

If image of closure belongs to closure of image, how to show preimage of interior belongs to interior of the preimage?

Here is exactly what I mean: Define a function $f:X\rightarrow Y$ from a metric space $X$ to another metric space $Y$. If any subset $A$ of $X$ satisfies $f(\bar A)\subset \overline {f(A)}$, then for ...
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0answers
65 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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0answers
180 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. ...
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1answer
102 views

A question about open balls in Hilbert space.

Let $S$ be a finite dimensional Euclidean space and let $B$ be an open ball of $S$. If $f$ is any homeomorphism of $S$ onto itself, then (it is easy to see that) $f(B)$ is a bounded and connected open ...
4
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1answer
159 views

Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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0answers
63 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
2
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0answers
161 views

Homeomorphism of compact Hausdorff spaces

In the preprint "A REMARK ON CANTOR DERIVATIVE" (http://arxiv.org/pdf/1104.0287v1.pdf), there is the next proof: We show that two countable locally compact Hausdorff spaces $X$ and $Y$ of same ...
2
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1answer
109 views

Extension of a function of a sphere to the disk.

Let $f:S^{n-1}\rightarrow Y$ be a continuous map from the sphere to a topological space. Why does $f$ have to be nullhomotopic for it to be extendable to the disk? I know this may be a silly question ...
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2answers
475 views

sequential continuity vs. continuity

A short and hopefully simple question for someone with more experience in topology: If a topology is induced by a mode of convergence and in fact nothing more is known apriori, whether this ...
3
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2answers
126 views

uniquness of limit of convergent sequence

I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. is there any space the limit of the convergence of sequence is not unique? -Thanks.
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0answers
45 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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1answer
68 views

Deleting a subset from a topological manifold

Consider an $n$-topological manifold $M$. We remove a subset $A$ from $M$. Are there cases where $M-A$ is no longer a topological manifold. In case we suppose that $M-A$ is still a manifold, what ...
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1answer
346 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let ...
3
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3answers
138 views

increasing sequence in $\omega_1$

I want to prove that a (countable) increasing sequence in $\omega_1$ converges to some point. Here is my idea: I first try to find the limit point. We know any point in the increasing sequence (say ...
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2answers
141 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
5
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1answer
124 views

Is continuous map from covering space to itself homeomorphism assumed both cover and base path-connected and $pf=p$?

In my topology assignment I came across the following problem: True or false? Let $E$ and $X$ be path-connected. For every covering map $p:E\rightarrow X$ and continuous map $f:E\rightarrow E$ ...
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1answer
47 views

why is successor ordinal open in omega_1?

I saw some arguments from the following link, Prove there are uncountably many open singletons in $ω_1$ But, I still can not figure out why it is open. What kind of open interval is it located in? ...
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1answer
45 views

an equivalent statement of “morphisms of projective varieties are closed”

I am interested in seeing why the statement (1) "If $Y$ is any variety and $Z$ a closed subset of $\mathbb{P}^n \times Y$, then the projection of $Z$ on $Y$ is closed." implies the statement ...
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1answer
31 views

How to show that $X$ has property $S_1(\mathcal O,\mathcal O)$, if and only if $\Sigma_{n=1}^{\infty} X_n$ does.

We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ \mathcal U_n : n \in \mathbb N \}$, there exists a sequence of open sets $\{ U_n : U_n \in ...
3
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1answer
81 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
2
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1answer
91 views

Why the space of probability measures is a subset of the measure space

Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
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1answer
63 views

Closure of $[1, 2] \cup [4, 5]$

The closure of a set is the intersection of all closed sets in $X$ which contain the set. In this the case of $[1, 2] \cup [4, 5]$ it seems to me that the closure will be $[1, 5]$, is that correct?
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1answer
32 views

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable. It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
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1answer
61 views

Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
1
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1answer
178 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
3
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1answer
43 views

Asymptotic cones of reals

Let me begin with the definition. Suppose $u$ is a free ultrafilter on $\omega$. Theorem. If $(r_n)$ is a bounded sequence of real numbers, then there exists a unique $l\in\mathbb R$ such that ...
4
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1answer
68 views

$A\cup B$ is connected when $A$ is connected in $X$ and $B$ clopen in $X-A$

Let $A$ be connected subset of a connected space $X$, and $B\subset X-A$ be an open and closed set in the topology of the subspace $X-A$ of the space $X$. Prove that $A\cup B$ is connected. I think ...
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2answers
64 views

How do I show that the map $p: X \to Y$, defined below, is a closed map?

$X = [0, 1] \cup [2, 3]$ and $Y=[0, 2]$ are subspaces of $\mathbb{R}.$ The map $p: X \to Y$ is defined as $$p(x)= \begin {cases} x&\text{if }x \in [0, 1]\\ x-1\ &\text{if }x \in [2,3]\\ \end ...
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0answers
31 views

Name for point in an infinite product of intervals bounded away from the boundary

Consider $[0,1]^{\omega}$ (the product of a countable amount of $[0,1]$ intervals). I am interested in points $x=(x_1,x_2,\ldots)\in [0,1]^{\omega}$ that are bounded away from the boundary, i.e. ...
2
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1answer
394 views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
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1answer
91 views

Universal cover of the pinched sphere?

Consider the sphere $S^2$ and identify its north and south poles to get a "pinched" sphere. What is the universal cover of this space?
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2answers
247 views

several questions about convergence in topological space

In a topology class, we define convergence as follow, a sequence $(x_n)$ converges to $p$ if for any open set $U$ containing $p$, we can always find an $N$ such that whenever $n \geq N$, $x_n$ is in ...
2
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1answer
64 views

Equivalence of continuous functions

Consider two topological spaces, $X$ and $Y$, and two continuous functions $f$ and $g$. By definition, given an open set $S$ in $Y$, the pre-image of $S$ under $f$ (or $g$) is an open set of $X$. Let ...
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2answers
106 views

on Continuous and Open Functions

Let $X,Y$ be compact Hausdorff spaces. If $f$ is a continuous function from $X$ onto $Y$, then $f$ is open. I am asking can the above result be proved. I am aware of the following cases: If $f$ ...
4
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1answer
94 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
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1answer
268 views

Prove that the a modified Cantor Set is not Jordan-Measurable

Let $C_0 = [0,1]$ and if $C_n$ is given as a disjoint union of intervals, construct $C_{n+1}$ by removing from each interval $I$ an open interval of length $(n+2)^{-2}|I|$ in the middle of each ...
2
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0answers
48 views

Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
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1answer
54 views

The topology of the restriction of a metric is the restriction of the topology.

I'm reading this proof of the following claim. Let $(X,d)$ be a metric space and $(Y,d')$ a subspace of $X$. If $(X,T)$ is the topology induced by $d$ and $(Y,T')$ the typology induced by $d'$, ...