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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How to describe this relation?

Let $\text{ann}(a;r_1,r_2)$ denote an annulus in $\mathbb{R}^2$ with center at $a$ and smaller radius $r_1$ and larger radius $r_2$, and $D_1(a,r_1)$ denotes the corresponding smaller disk. If there ...
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86 views

A question on Aut$(N)$ and Aut$(N/G)$

Let $N$ be a complex manifold and $G$ is a finite group freely acting on $N$. Define another complex manifold $M$ as $M=N/G$. I would like to study Aut$(M)$, the (holomorphic) automorphism group of ...
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95 views

‎sequential ‎space

A‎‎ ‎sequential ‎space ‎has ‎unique ‎sequential ‎limits ‎iff ‎each ‎countably ‎compact ‎subset ‎is ‎closed. ‎Proof: ‎If‎ ‎$ \{ x_n \} $ ‎is a‎ ‎sequence ‎converging ‎to ‎two ‎distinc ‎‎$‎x‎$ ‎and ...
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162 views

closed epigraphs equivalence

Is there a way to prove that the epigraph of any real function $f$ is closed iff $f$ is lower semi-continuous without using limit superior or inferior?
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52 views

Let $X$ be a $US$ space. Then $X^*$ is $US$ iff in $X$ , every convergent sequence has a relatively compact subsequence

$( X^*,\tau^*)$ is one - point compatification of topological space $ (X, \tau)$. A topological space is called a $US$-space provided that each convergent sequence has a unique limit. The bellow ...
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103 views

About $ \{ x \in[0,1]^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}$

Take $X$ a Tychonoff product $[0,1]^{\omega_1}$ and as $Y$ the $\Sigma$-product $$ \{ x ∈[0,1]^{\omega_1}:|\{\alpha<\omega_{1} :x(\alpha)\ne 0 \}|\le\omega \}\;.$$ The space $X$ is compact by ...
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22 views

A sequential minimal KC-space is compact

A space $(X,\tau )$ is said to be minimal KC , if $(X,\tau )$ is KC but no topology on X which is strictly smaller than $ \tau$ will be KC Theorem : A sequential minimal KC-space is compact. ...
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165 views

If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ Lindelöf

I have 2 different questions: As we know a space Y is Lindelöf if each open covering contains a countable subcovering. (1) :If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ ...
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52 views

Accumulation point

Accumulation point: Let $ \tau$ be a topological on $ ‎\mathbb{N}‎ $‎‎ such that is generated by $\{1,2\}, \{3,4\},\{5,6\}.... $. Let $A$ be non-empty of $ ‎\mathbb{N}‎ $‎‎ and $ n_{0} \in A$. If $ ...
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29 views

A hereditarily Lindelöf $KC$-space $( X,τ )$ is Katětov-$KC$ if and only if there is a weaker sequential $US$ topology $σ⊂τ

A space $( X,τ )$ is said to be Katětov $ KC $ if there is a topology $ σ⊂τ$ such that $( X,σ )$ is minimal $ KC $. The notion of strongly KC-spaces, that is, those spaces in which every ...
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141 views

A topological space that can be written as a countable union of pairwise disjoint clopen sets

Which topological spaces can be written as a countable union of pairwise disjoint clopen subsets? 1.One of them is the Baire space: $\omega ^{\omega} = [0] \cup [1] \cup ....$
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34 views

An infinite minimal strongly KC-space possesses a non-trivial

The notion of strongly KC-spaces mean spaces in which every countably compact subset is closed. a space $(X,‎\tau‎ )$ is said to be minimal strongly KC if $(X,‎\tau‎ )$ is strongly KC but no ...
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89 views

Sufficient conditions for $\pi_1(X\vee Y)=\pi_1(X)\ast\pi_1(Y)$

What are sufficient conditions such that $\pi_1(X\vee Y)=\pi_1(X)\ast\pi_1(Y)$ for spaces $X$ and $Y$?
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147 views

The algebra of clopen sets vs. the algebra of connected components

Let $X$ be a topological space which, for my intents, may be assumed to be metrizable and compact if needed (let's say it's a closed subset of the unit cube or something like that). I know that: If ...
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157 views

Loop spaces and filtered colimits

I have read many things now that lead me to believe that the loop space functor preserves filtered (and/or directed) colimits. Is this true? And can somebody give a (sketch of a) proof or point me in ...
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90 views

if $ X$ is a countable, compact $ T_{1} $ space and $ A ‎\subseteq‎‎‎‎ X $ then either $A$ is compact or…

Theorem: if $ X$ is a countable, compact $ T_{1} $ space and $ A ‎\subseteq‎‎‎‎ X $ then either $A$ is compact or there is a sequence in $A$ converging to point of $ X- A $. proof: Suppose ...
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67 views

Hausdorff space

we know that each infinite subspace of a KC space contain an infinite discrete subspace The Hausdorff spaces imply KC spaces.But: Is it right to say that every Hausdorff space has an infinite ...
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151 views

Continuity and product topology

I am not sure about the following. Let us consider the space of positive bounded sequences and a functional $(x_i)\rightarrow \sum_{i=1}^{\infty}\beta(i) x_i$, where $\beta$ is a function $\beta: N ...
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61 views

A non-compact $ T_{2} $ space which is not a k - space

A topological space is called k - space provided it has the property that any subset $ S $ such that $ S \cap K $ is closed for all closed compact $ K $ is itself closed. A topological space is ...
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69 views

Homotopy extension property and contractibility

Definition. A pair $(X,A)$ of topological space $X$ and its subspace $A$ satisfies Borsuk property if for any topological space $Y$ and for any continuous map $f \colon X \to Y$ any homotopy $F_A ...
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119 views

Help understanding CW-complex construction.

From p.5 of Hatcher's Algebraic Topology: I thought attachment maps would be from $n$-cell to $n$-cell since it says "by attaching $n$-cells", but in that link they're from $S^{n-1}$ to ...
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124 views

A weak version of Markov-Kakutani fixed point theorem

Let $\emptyset \not = X\subseteq \Bbb{R}^n$ be convex and compact and let $\cal{A}$ be a commuting family of affine maps from $\Bbb{R}^n$ into $\Bbb{R}^n$ such that $X$ is invariant under each element ...
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228 views

Alternative definition of covering spaces.

in a lecture I have seen a definition of a covering space, different from what I would call the usual one (e.g. the one in Munkres): A surjective continuous map $p:E\rightarrow B$ of spaces $E$ and ...
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74 views

Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
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62 views

Constructing a particular type of ultrafilter.

Let $u$ be a free ultrafilter on $\omega$. I am interested in constructing an ultrafilters on the closed subsets of $\mathbb R$ which contain the collection $\mathcal C$$=${$\bigcup_{n\in A} ...
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34 views

Convergence of a sequence in a nested local base

I am following a proof of the proposition: Let $B_{q} = \{B_{1},B_{2},...\} $ be a nested local base at $q \in X$ (X is a topological space) and let be a sequence such that $c_{1} \in B_{1}, c_{2} ...
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217 views

Stone-Weierstrass Theorem exercise.

Well, this is the exercise: Let $E,F$ be two compact metric spaces and $f:E\times F \to \mathbb{R}$ a continuous function. Show that for $\varepsilon >0$, exists a finite system ...
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2k views

What is the difference between a discrete function and a continuous function

Intuitively it seems that both concepts should be disjoint because if a function is discrete then it has some holes on it and if a function is continuous then it doesn't have holes. But now I'm not ...
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61 views

Weak star limit in $L^{\infty}(0,T;L^2(\Omega))$

Suppose that you know that $v_N$ is such that : $\forall N\in \mathbb{N}$, $v_N \in \mathcal{C}^0(0,T;L^2(\Omega))$ and $\partial_t v_N \in L^{\infty}(0,T;L^2(\Omega))$ ($\Omega$ an open bounded ...
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51 views

Question on boundedness

Page number 479 in partial differential equation by Evans book how to say that the derivative of I is bounded on bounded sets
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43 views

Family of maps from inverse system to another space

The universal property of an inverse limit $\lim_{\leftarrow} X_\alpha$ allows one to define from a compatible system of maps $\psi_\alpha: Y \rightarrow X_\alpha$ a unique map $\psi: Y \rightarrow ...
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132 views

helix and covering space of the unit circle

Does a bounded helix; for instance $\{(\cos 2\pi t, \sin 2\pi t, t); -5\leq t\leq5\}$ in $\mathbb R^3$ with the projection map $(x,y,z)\mapsto (x,y)$ form a covering space for the unit circle ...
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118 views

Proving that the map from a topological space to a fiber bundle is open.

There are two pieces to a question, the first: Let G be a group acting on a space $E$. Show that the quotient map $E \rightarrow E/G$ is open. The second: Let p : E → X be a fiber bundle. Show that ...
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48 views

Left and right homotopies have the same equivalence relations

Let $f, g \colon A \longrightarrow X$ be continuous functions of topological spaces. We say that $f,g$ are left-homotopic if there exists a continuous map $H \colon A \times I \longrightarrow X$ such ...
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230 views

Discrete topology on infinite sets [duplicate]

I want to prove the following: Let $X$ be an infinite set and $\tau$ a topology on $X$. If every infinite subset of $X$ is in $\tau$, then $\tau$ is the discrete topology on $X$. Proof. Let $x\in X$. ...
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47 views

How to build an specific atlas for the $n$-disk $D^n$

How can we build a topological atlas for the $n$-dimensional disk $D^n=\{ x\in \mathbb{R}^n \mid \lVert x \rVert \leqslant 1 \}$ as a manifold with boundary ? Specifically, how to construct the maps ...
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71 views

Potential function & constant

$\mathbf{Question:}$ Give a piar functions $Φ:\Bbb R^2 \to \Bbb R $ and $Ψ: \Bbb R^2 \to \Bbb R$, it is often useful to known that there exists some contiunously differentiable function $f:\Bbb R^2 ...
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77 views

How can I find a subbasis of a topology which does not contain a singleton set?

How should I prove this? Let (X,T) be a topological space. Find a subbasis S for T which does not contain any singleton sets. a. if X is finite. B. if X is infinite?
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62 views

Irreducibility of a topological space

Let $X$ be a topological space and $X=X_1 \cup X_2$ with $X_1, X_2$ nonempty open irreducible subsets. Then $X$ is irreducible iff $X_1 \cap X_2 \ne \emptyset$. The easy part: if it were $X_1 ...
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79 views

Open in topological spaces and the neighbourhood property

In metric spaces we know that every open set is one that has for each point a small neighbourhood that belongs to the set. But is this still true if we are talking about topological spaces? I am not ...
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98 views

Closed set in $l^1$ space

Let $$ X := \left \{ (a_n) : \sum_{n=0}^\infty |a_n| < \infty \right\}$$ with the metric $d(a_n,b_n) := \sum_n |a_n-b_n|$. Let $\delta_j^{(n)} := 1$ if $n = j$ and $0$ otherwise. Denote ...
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90 views

Question related to first order partial derivatives

If The funtion $f: \Bbb R^2 \to \Bbb R$ has directional derivatives in all directions at each point in $\Bbb R^2$ then the function $f$ has first order partail derivatives at each point in $\Bbb R^2$ ...
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41 views

Complete Linearly ordered Metric Space

Is there any Complete metric space other than $R$ or subsets of $R$ which is linearly ordered ?
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100 views

Is there a topology with respect to which a poset is connected iff its topologically connected?

A poset $P$ is said to be connected iff for all $x,y \in P$ we can 'zigzag' from $x$ to $y$ in finitely many steps. That is, iff for all $x,y \in P$ there exists a finite sequence $a : n \rightarrow ...
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45 views

Is convergence in $I^I$ topology equivalent to point-wise convergence?

The $I^I$ topology is the uncountable Cartesian product (Tychonoff) of the closed unit interval $[0,1]$. We can imagine it as a space of all the functions from $[0,1]$ to itself. I was told that a ...
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98 views

The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable

Let $X$ be a Banach space. If $B\subset X$* is a norm-separable How can we prove that: The topology on bounded sets in $X$** of pointwise convergence on $B$ is metrizable. $X$*$=B(X,\mathbb{R})$ : ...
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320 views

Why is the inverse image of a compact set under a special sort of function compact?

Let $f$ be a continuous closed function from $X$ to $Y$ where $X$ and $Y$ are topological spaces. (Closed means that for any closed set $C$, $f(C)$ is also closed). Suppose that for any $y$ in $Y$, ...
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309 views

multiple choice question on topology of matrices

Which of the following are true? a)The set of symmetric positive definite matrices are connected. b)The closed unit ball centred at $0$ and of radius $1$ of $l_1$ is compact c)The set of invertible ...
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114 views

Prove that $ \bigcap _{i=1}X_i$ is connected. [duplicate]

Let $X$ be a compact Hausdorff space and let $X_1 \subset X_2 \subset X_3 \subset \cdots$ be a sequence of closed, connected subspaces. Prove that $\bigcap_{i=1}^\infty X_i$ is connected. Give an ...
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49 views

Extending Coverings on dense subsets

Let $X$ be a metric space with $D \subseteq X$ a dense subset. If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$? ...