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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270 views

Regular covering implies transitive automorphism group action

We already know the theorem Theorem Let $p: (Y,y) \rightarrow (X,x)$ be a covering, with $Y$ connected and $X$ locally path connected, and let $p(y) = x$. If $p_*(\pi_1(Y,y))$ is a normal ...
2
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1answer
372 views

On the generalized Sierpinski space

In Sierpiński topology the open sets are linearly ordered by set inclusion, i.e. If $S=\{0,1\}$, then the Sierpiński topology on $S$ is the collection $\{ϕ,\{1\},\{0,1\} \}$ such that ...
2
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1answer
78 views

Union of compact sets in a convergence space

Let $X$ be a convergence space and let $K_1, K_2, \ldots, K_n$ be compact subsets of $X$. I'm trying to prove for myself that the union $K$ of the $K_i$ is compact. By definition, $K$ is compact if ...
2
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1answer
655 views

Let $X = \Bbb{R}$ with the discrete metric. Is $X$ connected?

No. Any nonempty subset $A ≠ X$ is open, as well as its complement. So $X$ is the union of disjoint nonempty open subsets. Is there a more formal way of doing it? Thanks for your help.
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1answer
140 views

Weak a.s. convergence VS a.s.weak convergence

Let's consider a sequence $(\mu_n)_n$ of random probability measures on $\mathbb R$, and let $C_b$ be the Banach space of bounded continuous functions on $\mathbb R$. I am considering the following ...
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1answer
151 views

Limit on a topological vector space

in the Wikipedia article on Gâteaux derivative , the limit of a function between two topological vector spaces is taken. How is the limit defined on a topological space for a function ? I find ...
2
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1answer
192 views

Flabby sheaves and comparison of topologies

Let $A^p$ be a group of sheaves on a topological space $X$, let $F$ be the global sections functor $F(A^p) = A^p(X)$. I have to compute the cohomology of the complex $0\rightarrow A^1(X) \rightarrow ...
2
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1answer
122 views

technical question about covering maps

I have been learning about covering maps, and I am having trouble proving something which my intuition is telling me should be true. For this question, a covering map is a continuous, surjective map ...
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1answer
169 views

Bigon question related to Dehn twists

Perhaps someone can help me with this: For simple closed curves on an orientable compact surface, if they form a bigon, then is it true that at the intersection points the orientations must be ...
2
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2answers
89 views

Can the borders of a map be deformed to give arbitrary area to any region?

Let's say I have a geographic map, a connected region divided into sub-regions. Is it possible to deform the map (the borders of the regions) so that each sub-region is of arbitrary area while ...
2
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1answer
120 views

How to show that the identity is in closure of $O' \subset C(\mathbb R)$?

The sets $\{f \in C(X) : |g-f| \leq u \}$ where $g\in C(X)$, $u$ a positive unit of $C(X)$ form a base for some topology on $C(X)$. Let $X = \mathbb{R}$, the set of real numbers. With the above ...
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108 views

Almost invariant subspaces for WOT closure of an algebra of operators

Let $X$ be a Banach space and $\mathcal{C}\subset\mathcal{L}(X)$ be a collection of bounded linear operators. A subspace $Y$ is said to be almost invariant under $\mathcal{C}$ if for each ...
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1answer
74 views

Quotients and continuous maps

Suppose $f:\mathbb R\times \mathbb R\to \mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$ where the domain is given the usual topology and the latter the quotient topology. Why then is the restriction ...
2
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1answer
173 views

Irreducible set

Let $X$ be a topological space Let $A$ be the set of all closed, irreducible subsets of $X$ equipped with a topology that contains all sets of the form $V(U)=\{a\in A| a\cap U\neq\emptyset, ...
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1answer
114 views

Product of Transitive Systems

Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$. We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
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1answer
303 views

Box topology on $\prod_{n=1}^\infty\mathbb{R}$

Let $X$ denote $\prod_{n=1}^\infty\mathbb{R}$, the Cartesian product of countably infinitely many copies of $\mathbb R$ (which is just the set of all infinite sequences of real numbers), endowed ...
2
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1answer
77 views

Is there an unbiased random walk on a colored plane for any number of colors?

So I was trying to motivate the fundamental postulate of statistical mechanics (i.e. all microstates are assumed to be equally probable $-$ even if we can't practically measure them, but only their ...
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1answer
124 views

At what point or points does the family $\{f_n\}$, where $f_n(x)=x^n$, fail to be equicontinuous?

I am currently learning topology from Munkres. The question below is an exercise in section 45. Let $f_n\colon I\to \mathbb{R}$ be the function $f_n(x)=x^n$. The collection $F=\{f_n\}$ is ...
2
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1answer
166 views

Each discrete space is a Polish Space

I'm trying to solve exercise 6.3#7 from Sidney A. Morris' "Topology without tears": "Prove that each discrete space [...] is a Polish space." I started by proving that discrete spaces are always ...
2
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1answer
117 views

How to preserve completeness between different metrics on the same space?

Let $(M,d)$ be a metric space and $f\colon[0,\infty)\to[0,\infty)$ metric preseving map that is right continuous at $0$, i.e. $f$ has satisfies $$\forall x,y\in [0,\infty)\colon f(x+y)\le ...
2
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1answer
230 views

Show that each component of $X = \{a, b, y_1, y_2, \ldots\}$ is Hausdorff, but $X$ is not Hausdorff.

Given $X = \{a, b, y_1, y_2, \ldots\}$, where $a$, $b$, $y_i$ are points. The topology on $X$ is $\tau = \{X\} \cup T_f\cup T_d$, where $T_f$ is the cofinite topology on $X$ and $T_d$ is the discrete ...
2
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2answers
217 views

Showing complement of subsets are open

How do you show the complement of these subsets are open? $S^1=\{(x_1,x_2)\in\Bbb R^2\mid x^2_1+x^2_2=1\}$ $B^c_1=\{(x_1,x_2)\in\Bbb R^2\mid x^2_1+x^2_2\ge1\}$
2
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1answer
303 views

why is the infinite product of the discrete two point space with itself, a topological homogeneous space?

Why is the infinite product of the discrete two point space with itself, a topological homogeneous space?
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1answer
162 views

Proof that no segment is compact

Does anyone have a simple proof (without using any theorems of compactness) that no segment of the form $(a, b)$ in $\mathbb{R}$ is compact? Definitions: For any subset $E$ of a metric space $X$, an ...
2
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1answer
265 views

separation properties in Hausdorff, compact spaces

Suppose $X$ is a compact Hausdorff topological space, $C\subseteq X$ a closed subset and $x\notin C$ a point. I have to prove that there exists a compact neighborhood of $x$ which is disjoint from ...
2
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1answer
84 views

If $x_n\to \overline{A}$ does it mean that $x_n\to A$?

Let $(X,d)$ be a metric space. The sequence $(x_n)$ converge to the set $A\subset X$ (denoted as $x_n\to A$) iff $$ \lim\limits_n d(x_n,A) = 0 $$ where $d(x;A) = \inf\limits_{y\in A}d(x,y)$. Let ...
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1answer
51 views

On continuity of a binary map between arbitrary topological spaces

Let $X$, $Y$, $Z$ be arbitrary topological spaces and $f:X\times Y \rightarrow Z$ be an arbitrary map. Then, is it true that $f$ is continuous iff for every $y \in Y$ $f(\cdot,y):X\rightarrow Z$ is ...
2
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1answer
144 views

Generalizations of the Convex Hull

I am aware of generalizations of the convex kernel, via the addition of more polygonal line segments between points in a set. However, I wonder if there are similar generalizations for the convex ...
2
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1answer
422 views

Projective space as a manifold

In the projective space $\mathbb{P}^n(\mathbb{K})$ I can consider the hyerplane $H_{0}=\{x_{0}=0\}$ and the set $U_{0}=\mathbb{P}^n(\mathbb{K})-H_{0}$. Clearly $\mathbb{P}^n(\mathbb{K})=U_{0}\cup ...
2
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1answer
184 views

Alternate definition of closed set in $\mathbb{R}$

Show that a subset of $\mathbb{R}$ is closed iff it contains all its accumulation points. Well, the definition of accumulation point for a set S is that I have is that for all $\epsilon>0$, ...
2
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2answers
394 views

Locally compact topological group is Normal

How can I prove directly that a locally compact topological group G is normal? I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove ...
2
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1answer
150 views

Connected subsets of lines

For me every connected set has at least two elements. Connected subsets of the real line have non-empty interiors. I am curious if the same is true for connected subsets of any connected linearly ...
2
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3answers
273 views

reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
2
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1answer
178 views

Critical points of a proper holomorphic map

This should probably be easy, but I can't prove it. Let $f : U^{\prime} \rightarrow U$ be a holomorphic, proper map of degree $d$. Here, $U^{\prime}, U \subseteq \mathbb{C}$ are open sets, both ...
2
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1answer
160 views

Existence of a structure-preserving mapping between two spaces?

I have some questions, but not sure if they are meaningful: Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$? Suppose $X$ and $Y$ are ...
2
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1answer
117 views

weakly locally one-to-one?

Is there any standard name for this concept that is weaker than local one-to-one-ness? In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$. Or, if you like: In ...
2
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1answer
249 views

order topology vs product topology for an infinite product

Let $A$ be a finite set with the discrete topology (i.e. every subset is open), and $\mathbb{N}$ denote the set of natural numbers. Is the product topology on $A^\mathbb{N}$ equivalent to the order ...
2
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2answers
234 views

Sequential continuity for quotient spaces

Sequential continuity is equivalent to continuity in a first countable space $X$. Look at the quotient projection $g:X\to Y$ to the space of equivalence classes of an equivalence relation with the ...
2
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1answer
129 views

starting with paths and the fundamental group

I have two problems, I´m a little complicated with the problem , I know that it´s easy but I need just a little help. Are two problems i) Suppose that the identity map $ i:X \to X $ is homotopic to ...
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1answer
58 views

Follow-up: that the embedding is the intersection of closed sets

I was reading through general-topology posts and I couldn't quite understand this one. I tried asking directly on the thread, but I didn't get a response. This is in reference to Professor Israel's ...
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1answer
236 views

Why is this embedding a homeomorphism?

I want to ask a follow up question to the one I asked earlier here. In Robert Israel's answer, it's posited that the natural embedding $\iota$ of $B = \{y \in \ell_\infty: \|y\| \le 1\}$ into $P = ...
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2answers
209 views

Notation of a subset

Is it meaningful to write $U\subset(X,\tau_X)$ where $(X,\tau_X)$ denotes a topological space? Or is it better to write $U\subset X$? Or in fact $(U,\tau_U)\subset (X,\tau_X)$? Thanks!
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1answer
583 views

Topology: Proving a space is connected

I'm attempting to prove that a space is connected and compact. I have a continuous function $f:X \rightarrow S^{1}$. $X$ is metrizable and locally connected. $f$ is non-constant, surjective and ...
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2answers
120 views

Topologies on spaces of mappings

Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that ...
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1answer
365 views

Principal and fiber bundles as defined by Husemoller

In his book 'Fiber Bundles' Husemoller defines principal bundles and fiber bundles quite differently from how they are usually defined. Specifically: Definition: a right $G$-space $X$ is called ...
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2answers
115 views

Unsure lines in a proof of Tychonoff's Theorem

Can someone explain some of the later lines in this proof? I think $\mathcal{T}$ is meant to be $\mathcal{T_i}$ in the third to last line. After that, how is one sure that $x$ can be found in some ...
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1answer
496 views

Complicated with the quotient topology (Möbius strip) two distinct quotients

Let follow this definition of manifold. An n-manifold is a Hausdorff Topological space, Such That Each point you have an open neighborhood homeomorphic to the open disc $$ U^n = \left\{ {x \in R^n ...
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2answers
135 views

3D surface and topology

I was reading an article that mentioned "a connected surface in 3D space with $\infty$ many ends (in the topologocal sense)". I have read the wiki page on "ends" but couldn't make much sense of it, ...
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1answer
256 views

Genus-g surface with 2 boundary components and non-trivial (non-bounding ) curves

Let $S_{g,2}$ be the orientable genus-$g$ surface with two boundary components, and let C be a simple-closed curve in $S_{g,2}$. If C is homologically non-trivial (i.e., C does not bound a surface), ...
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2answers
1k views

products of closure, is the closure of the product

This is part of the proof of Munkres Book. Conversely, suppose $ $$ x = \left( {x_\alpha } \right) $$ $ lies in the closure of $ $ , in either topology ( box or product). We show that for any ...