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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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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62 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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109 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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40 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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88 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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70 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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76 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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40 views

Is there a name for a set where any two elements are separated by a given distance?

I am curious if there is a name for such a set. Let $(M,d)$ be a metric space and $S$ a subset of $M$ for which there is some positive number $\delta$ such that for any two distinct elements in $S$, ...
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92 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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56 views

What can we deduce about $A$ and $B$ if $\bar{A} \cap \bar{B} = \bar{A}$?

I suddenly came across with the following problem (or exercise?) in topology that I can not get out of my mind. Let $A$ and $B$ be non-closed subsets of a (non-discrete) topological space $X$. ...
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38 views

the compact Hausdorff $\beta X$

The bellow example show that the compact Hausdorff $\beta X$ does not have non-trivial subsequence: Assume $R=(\mathcal{U}_n)_n$ is a sequence of distinct ultrafilters on some set $X$. Since every ...
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173 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
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45 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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56 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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84 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 ...
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29 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector ...
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69 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
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37 views

Does there exists a simple imbedding theorem for general topological $n$-manifolds?

I am interested in finding some paper or book where i can find how to build an imbedding $e\colon M^n \hookrightarrow \mathbb{R}^q$ of an arbitrary Hausdorff topological $n$-dimensional manifold $M$ ...
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144 views

Proving a set is compact?

So let || || be any norm on $\mathbb{R}^m$ and let $B = \{ x \in \mathbb{R}^m:$ || $x$ || $\leq 1 \}$. I want to prove that B is compact. So from the Heine-Borel Theorem, I know that every closed and ...
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63 views

inversion on $\mathbb{R}^n/\{0\}$ is homeomorphism

Question is to prove that : Inversion on $\mathbb{R}^n/\{0\}$ is homeomorphism. By inversion i mean for a fixed real number $R>0$ the function $f : \mathbb{R}^n/\{0\}\rightarrow ...
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63 views

Trying to show some spaces are homeomorphic

I've been given the definition that $X$ and $Y$ are homomorphic if there is a bijection between them, with both the function and inverse being continuous. Haven't really looked at them other than ...
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88 views

Help, please! trigonometry and linear system

I have this exercise; can anyone help me please? $\def\arsinh{\operatorname{arsinh}}$ We have $$\forall x,x' \in \mathbb{R}, |\tanh (x) - \tanh (x')| \leq |x-x'|$$ and $$\forall x,x' \in ...
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74 views

$U,V\subseteq X^\omega$ and X is w.o. such that $U'\subseteq U$ and $V'\subseteq V$ open and $U'\cap V'=\emptyset$ and $U'\cup V'=U\cup V$

Suppose $U,V\subseteq X^\omega$ (where $X^\omega:=\{f| f:\omega\to X\}$) and X is wellorderable, how it would be possible to show that there are open sets $U'\subseteq U$ and $V'\subseteq V$ such that ...
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101 views

Equivalent statements about linear functionals in connection with the topology which we can obtain from seminorms

I have got a question about the topology we can get from out a separeting family of seminorms (to make a topological vector space from out an arbitrary vector space). There exists the following ...
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42 views

Group of covering transformations

The group of automorphisms of a covering $p: E \mapsto X$, to be denoted $Aut(E,p)$, is usually referred to as the group of covering transformations. If $p: E_1 \mapsto E_2$ is an isomorphism of ...
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250 views

Complete but not totally bounded metric space with a certain property

Give an example of a complete metric space $(X,d)$ and a nested sequence of nonempty closed BALLS $A_n = \bar{B}(x_n,r) = \{y \in X : d(x_n,y) \leq r\}$ such that $\bigcap_n A_n = \emptyset$. So ...
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65 views

Miranda's book pag. 60

The author’s dealing with the connected sum of two topological spaces. He defines the map $\pi: X \sqcup Y \mapsto Z$ and endows $Z$ with the quotient topology. He also says that the natural ...
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71 views

understanding Poincare conjecture

I was reading a Wikipedia article on Poincare conjecture and found out that it basically says that if each loop in the space can be continuously tightened to a point then the space is topologically ...
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247 views

Is my proof for the translation of an f-sigma set valid?

Let $F = \cup^{\infty}_{k=1} A_i$ where each $A_i$ is a closed set. Since $F$ is $F_\sigma$, every $F_\sigma$ set is measurable, and every measurable set is translation invariant, F is translation ...
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75 views

Euclidean metric is equivalent to rectangular metric in $\mathbb{R}^2$

$FACT$: Let $X$ be a topological space with two bases $B$ and $B'$. Then $B$ is equivalent to $B'$ iff for all $N \in B$ and for all $x \in N$, there is $N' \in B'$ such that $x \in N' \subseteq N$ ...
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81 views

Literature leading up to algebraic topology

First and foremost, allow me to apologise for the quite general and perhaps vague nature of this question and for the presence of any misunderstandings in the statements I will use. I know very well ...
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79 views

Importance of Banach fixed point in mathematics

I know the proof of Banach fixed point theorem in complete metric space and two example of it. It is useful to prove Picard's theorem on existence and uniqueness of ordinary differential equation. In ...
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35 views

Is Hol(S,T) locally compact?

$S$ and $T$ are compact Riemann surfaces,$Hol(S,T)$ denotes all the holomorphic fuctions from $S$ to $T$,if we give the "locally uniform convergence" topology to $Hol(S,T)$,How to prove $Hol(S,T)$ is ...
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38 views

Is a subcompact space submetacompact?

Is a subcompact space submetacompact? The space is subcompact if it has a base $\mathcal B$ such that every regular filterbase $\mathcal U \subset \mathcal B$ has non-empty intersection. This ...
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97 views

$x$ is a limit point of $A\subseteq\mathbb{R}^n$ iff $\exists$ a sequence $\{x_i\}_{i\in\mathbb{N}}$ in $A$ so that $x$ is a limit point of $\{x_i\}$

PROBLEM: $x$ is a limit point of a set $A \subseteq \mathbb{R}^n$ iff $\exists$ a sequence of points $\{x_i\}_{i \in \mathbb{N}}$ where $x_i \in A$ so that $x$ is a limit point of $\{x_i\}$. My ...
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116 views

Continuous bijection between an annulus + a point and the open unit disk

The open annulus with a point I define as $\{ (r,\theta)\colon 1<r<2,0\leq \theta < 2\pi \}\bigcup \{\left(1,0\right)\}$. Call that $A$. Let $B=\{(r,\theta)\colon 0\leq r<1, 0\leq \theta ...
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56 views

Error in Lang's definition of weak topology?

On page 23-24 of his Real and Functional Analysis (3e) Serge Lang claims Let $Y$ be a topological space and let $\mathscr{F}$ be a family of mappings $f \colon X \to Y$ of $X$ into $Y$. Let ...
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48 views

JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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62 views

On Some Locally Convex Topologies of a Vector Space(Update)

This is an update of my previous question in here. Suppose that $(X,\tau)$ is already a locally convex TVS. Let us denote by $X'$, the space of all $\tau$-continuous linear functionals on $X$, the ...
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145 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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47 views

non trivial convergent sequence

Let $\beta\omega$ be the Stone-Čech compactification of the natural numbers. We know that it is compact and Hausdorff, but it has no non-trivial convergent sequence. Is there an example else to be ...
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91 views

Union of Sets in Locally Compact Hausdorff Space

Is it possible for an open set in a locally compact Hausdorff space to not be the union of an increasing sequence of compact sets? If so, given a regular Borel measure on such a space, how is it that ...
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77 views

sets in the classical cantor space and how do they look in $2^{\mathbb N}$?

Classical cantor = $C$ and $K = 2^{\mathbb{N}}$. $A_{n} = \left[\dfrac{1}{ 2^{2n-1}}, \dfrac{1}{2^{2n-2}}\right]\cap C$. Then $A_{n}$ in $K$ is ?
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135 views

Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
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79 views

Markov kernels and update functions

I would like to prove, that for any Markov kernel $K$ on a Polish space $(F,\mathcal{F})$ (with a $\sigma$-field) you can find a measurable space $(S,\mathcal{S})$, a random element $Z$ on $S$ and an ...
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53 views

When the $(\mathbb R,+)$-action defined by the flow of a vector field on a manifold is proper ?

I'm interested in the conditions that a vector field has to satisfy in order for its flow to define a proper action. Technically, let $\xi$ be a smooth vector field. For each $m\in M$, there is a ...
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65 views

Proof of an analogue of Lagrange's Theorem for profinite groups

I have been trying to work through a proof for Lagrange's theorem extended to profinite groups from John Wilson's book. The statement of the theorem is "for $H$ and $K$ subgroups of a profinite group ...
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50 views

Given an abstract open book obtaining a 3-manifold

Let $\Sigma_{\phi}$ be the mapping torus of $\phi$, i.e., $\Sigma \times [0,1] / \sim$ where $(\phi(x),0) \sim (x,1)$ for all $x \in \Sigma$. Also let $$M_{\phi} = \Sigma_{\phi} \cup_{\psi} ...
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47 views

some question in the proof of classification of compact connected surface

Each compact connected $2$-manifold $S$ has a proper triangulation $K$, so we can order all $2$-simplices of $S$, $F_1,F_2,\ldots,F_{k-2}$ such that $F_i$ meets $F_{i-1}\cup F_{i-2} \cup \ldots \cup ...
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91 views

How to turn a topological space into a semi-decidable logic?

In two interesting posts(here and here),it is mentioned that "there is a close connection between semi-decidable logics and topological spaces" Michael O’Connor wrote: In fact, given a ...