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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Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
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281 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
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146 views

Understanding Sierpinski carpet formally

In this paper, one definition of carpet(Sierpinski) is given as follows: A metrizable topological space is a carpet iff it is a planar continuum of topological dimension 1 that is locally connected ...
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71 views

How does the base in the complete lattice from a given topology look like?

I been reading back and forth about lattices and topologies all day. And one thing I can't seem to get a good idea about is how a base would look in such a complete lattice. And what would be the ...
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132 views

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. Prove that the image under $f$ of each interval is either a single point or an interval. (w/o IVT)

I'm reading Intro to Topology by Mendelson. The problem statement is in the title. Also, I'm in the section right before he introduces the Intermediate Value Theorem, so I want to try and not use it ...
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704 views

product of path connected space is path connected

Will product of path connected topological spaces be necessarily path connected? Why or why not? Give me some hints. Thank you.
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1answer
118 views

Counterexample or proof that a certain subset in a topological group is closed

We consider a Hausdorff topological group $G$ acting on a topological space $X$ [action simply means a continuous map $G\times X\rightarrow X$ verifying $(gh)(x)=g(h(x))$, and $1(x)=x$]. The set ...
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101 views

Completion of a metric space

I got a doubt with the next exercise. Let $(X,d)$ be a metric space. Denote $\mathcal{B}(X,\mathbb{R})$ the subset of all bounded functions from $X$ into $\mathbb{R}$. Let $a \in X$. Show that ...
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148 views

What is $S^3/\Gamma$?

Let G is a group and H is a subgroup of G. I know $G/H$ is the quotient space but I have no idea about what $S^3/\Gamma$ is, where $S^3$ is the sphere and $\Gamma$ is a finite subgroup of $SO(4)$. In ...
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154 views

How to define an interior point in terms of $\epsilon$-balls?

Which is the technically correct definition? I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$. II) An interior point of a set $B$ is a point that is in ...
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KC - space and FDS - property

I saw a question that was asked by " Maryam " she asked: Let $(X,\tau)$ be a $T_1$ topological space. Let $D=\{d_n:n\in\omega\}$ be a countably infinite closed discrete subspace of $X$. Fix $p\in X$ ...
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284 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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763 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
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38 views

Some doubts about the boundary of set $A$ in the ambient space $V$ (in the book: Multidimensional real analysis. 1)

When I reading Duistermaat J. & Kolk J's Multidimensional real analysis. 1 Differentiation.(you can read from here: Multidimensional real analysis. 1) In page 11. I can't understand the ...
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251 views

Every Borel set is the union of an increasing sequence of Bounded Borel sets?

I am currently working with the book by Halmos, and i can't quite get past this one. It states that: "Every Borel set can be written as an increasing sequence of Bounded Borel sets" In this case $X$ ...
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108 views

What is the algebra generated by a set of functions?

Well, consider $X$ a metric space, my main doubt is: What is the sub-algebra, of the algebra of continuous functions $\mathcal{C}(X,\mathbb{R})$, generated by a set $S$ of continuos functions? Also: ...
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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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85 views

Does a neighbourhood need to be a *connected* set?

I have in my topology/ real analysis textbook the definition of neighbourhood of a point as an open set containing that point. But isn't a neighbourhood necessarily a connected set? Wikipedia also ...
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414 views

continuous onto map from $(0,1)\to (0,1]$

I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$ could any one give me any hint?
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72 views

minimal KC - space and strongly minimal KC - space

If P is a topological property, then a space $(X, \tau)$ is said to be minimal $P$ (respectively, maximal) if $(X, \tau)$ has property $P$ but no topology on $X$ which is strictly smaller ...
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1answer
248 views

Fixed point of a continuous map

Take $X = [0,1]$, and a continuous map $f:X \rightarrow X$. Then there exist a point $x \in X$ s.t. $f(x) = x$. We may take $X = (0,1)$ or $X = (0,1]$. Shall we get such fixed points in latter cases? ...
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104 views

KC-spaces and US-spaces.

A topological space is called a US-space provided that each convergent sequence has a unique limit. A topological space is called a KC-space provided that every compact subset is closed. So ...
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85 views

topological KC - space

A topological space X is KC – space if every compact subsets are closed. question: Does a KC - space contains a minimal KC topology?
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69 views

minimal KC and (strongly) KC

If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...
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92 views

topological space

Let $( X,\tau )$ be a $T_1$ topological space. Let $D = \{ d_n : n \in \omega \}$ be a countably infinite closed discrete subspace of $X$. Fix $P \in X$ and let $F \in \beta\omega- \omega$ be an ...
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144 views

Orthogonal chords of compact sets

For any compact set on a plane say C does there always exist a chord in C such that its end points are orthogonal to the boundary of C (assumed smooth)
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279 views

Properties for interior and closure in metric space.

I found the some following properties for general topology and prove these. But, I want to verify that the proofs are really true. Let $(X,d)$ be metric space. Let $A$ be any subset of $X$. Define ...
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1answer
81 views

How to prove this version of Urysohn's Lemma using the usual version

I'm trying to prove the Urysohn's lemma that is presented in Rudin's Real and Complex Analysis using the usual version of the Urysohn's Lemma (with normal space ...). Here is the Rudin's Version: ...
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1answer
137 views

Is a single point boundaryless?

I am trying to understand Preimage orientation. So I got this question: Definition. The boundary of $X$, consists of those points that belong to the image of the boundary of $\mathbf{H}^k$, the ...
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Proving Every open set in $\Bbb R$ is a countable union of open intervals. [duplicate]

This question is from William R. Wade's Introduction to Analysis book: Prove that every open set in $\Bbb R$ is a countable union of open intervals. I have no ideas honestly. Thank you.
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592 views

Does a compact connected complete linear order have the fixed point property?

Would the same arguments used for showing $[0,1]$ has the fixed point property hold in this general case? What could go wrong? EDIT: The fixed point property can be interpreted in two ways that I ...
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82 views

Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ has a finite subcovering.

Let $H \subseteq \Bbb R^n$. Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ where $E_{\alpha}$'s are relatively open in $H$ has a finite subcovering. $\bf{Solution \ ...
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75 views

Prove that $H$ is a finite set.

Let $H$ be compact in $\Bbb R^n$ Also assume that for every $x\in H$ there is an $r=r(x)$ such that $B_r(x)\cap H=\{x\} $ Prove that $H$ is a finite set. Solution: Since $H$ is compact, ...
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1answer
68 views

Show that $Y$ is locally compact.

Recall that a space $X$ is locally compact if for any point $x$ in $X$, and any neighborhood $U$ of $x$ in $X$,there is a neighborhood $V$ of $x$ in $X$,and a compact subspace $C$ of $X$ such that $x ...
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105 views

Prove that f is constant on $K$ that is, if $a \in K$ then $f(x)=f(a) \ \ \forall x\in K$

Suppose that $f: \Bbb R^n \to \Bbb R^m$ and that $a\in K$, where $K$ is a compact connected subset of $\Bbb R^n$ suppose for each $x\in$ $K$, $\exists$ $\delta_x >0$ such that $f(x)=f(y)$ ...
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166 views

Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty.

Question: Let $E$ be a subset of $\Bbb R^n$ Prove that $a$ is a cluster point of $E$ $\iff$ for each $r>0$, $E\cap B_r(a)$ \ $\{a\}$ is nonempty. definiton: A point $a \in \Bbb R^n$ is ...
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270 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
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316 views

Prove that every closed ball in $\Bbb R^n$ is sequentially compact.

Question: Prove that every closed ball in $\Bbb R^n$ is sequentially compact. A subset $E$ of $\Bbb R^n$ is said to be squentially compact $\iff$ every sequence $x_k\in E$ has convergent ...
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122 views

Structure of topological spaces in terms of sequences

I been reading for several hours and not yet found a question put in this way. Given any topological space: Does every sequance in $X$ determine a countable subset of $X$? Do the sets that belong ...
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1answer
80 views

How to prove that initial arrows in Haus coincide with topological embeddings?

In Joy of Cats it is stated that in category $\textbf{Haus}$ initial arrows coincide with topological embeddings (pg 135). This can be proved by showing that initial arrows in $\textbf{Haus}$ are ...
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210 views

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets.

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets. I researched the meaning of "clopen set". And I reached the result that so as to ...
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2answers
100 views

domain of initial $f : X \rightarrow Y$ in Haus equipped with coarsest topology?

If $f:X\rightarrow Y$ is initial in category Top then it is easy to proof that (!) the topology on $X$ is the set of preimages of open sets in $Y$. Just construct topology $Z$ having the same ...
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45 views

Question on Computation of Integral of a Form

Again: I'm trying to understand the result of a certain integral of a form in a paper I'm reading (for which I do not, unfortunately, have a link): We start with a surface S that is oriented, ...
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2answers
86 views

Compact and countability axioms!

I wondering which countability axioms compact imply in arbitrary topological spaces. I'm using Greene/Gamelin 2nd ed. And they list separable, 2nd-coutable, first-countable and Lindelöf. Clearly ...
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62 views

On the proof of deformation lemma “boundedness”

Book- Evans partial differential equation. In the proof of deformation lemma how to say that $V(u)=-g(u)h(\lVert I'(u)\rVert)I'(u)$ is bounded. And how to say that the mapping $u \to ...
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167 views

What can we say about closed sets in the Baire space that are neither open nor compact?

I'm trying to figure out what closed subsets in $\omega^{\omega}$ equipped with product topology should look like. It seems to me it's relatively easy to have an idea about compact closed subsets and ...
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100 views

Checking for uniform convergence $f_{n}(x)$ $=$ $n^{2}x\ (1-x)^{n}$

Let $f_{n}(x)$ $=$ $n^{2}x\ (1-x)^{n}$ with $0\le x\le 1$, where $f_{n}$ converges pointwise to the zero function $f$. How do I check for uniform convergence? Can someone provide me with some hints? ...
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4answers
101 views

Closure of a certain subset in a compact topological group

Suppose that $G$ is a compact Hausdorff topological group and that $g\in G$. Consider the set $A=\{g^n : n=0,1,2,\ldots\}$ and let $\bar{A}$ denote the closure of $A$ in $G$. Is it true that ...
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1answer
53 views

Why do the author added the extra condition that $X$ needs to be $T_1?$

In my text it's written that, But I get to prove the result underlined red simply for a first countable space as: (N.B. by limit point the author wanted to mean the adherent point) ...
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1answer
55 views

Is there any way to prove it directly?

I'm trying to prove the following result: In a first countable $T_1$ space $X$ for $E\subset X,~x\in X$ is an adherent point of $E\iff~\exists~(x_n)_n\in E$ such that $x_n\to x.$ When I'm ...