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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Question on product measure

Suppose that $m$ is the standard product measure on $S=\{0,1\}^A, (A>\omega)$. Is there an open subset $U$ of $S$ such that $m(U)\neq m(\overline U)$, where $\overline U$ is the closure of ...
3
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0answers
15 views

Extending a Homeomorphism between open ball and open box of $R^n$

I am wondering if the following is true: Suppose $U$ is the unit open ball in $\mathbb{R}^n$, and $V = (0,1)^n \subset \mathbb{R}^n$. Suppose $h: U \to V$ is a homeomorphism. Then there exists a ...
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2answers
20 views

Extending a Homeomorphism between open ball and bounded open subset of $R^n$

I am wondering if the following is true: Suppose $U$ is the unit open ball in $\mathbb{R}^n$, and $V \subset \mathbb{R}^n$ are bounded open set. Suppose $h: U \to V$ is a homeomorphism. Then ...
2
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1answer
18 views

Extending a Homeomorphism between bounded open subspaces of $R^n$

I am wondering if the following is true: Suppose $U, V \subset \mathbb{R}^n$ are bounded open subsets. Suppose $h: U \to V$ is a homeomorphism. Then there exists a homeomorphism $H : \overline{U} ...
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25 views

If $A$ be a connected set in a metric space and $B\subseteq A^‎{'}$, which is the connected set? [on hold]

Let $A$ be a connected set in a metric space and $B\subseteq A^‎{'}$. which the follow set is the connected set? $A^{\circ}$ $\overline{A}-B$ $\overline{A}-A$ $A\cup B$
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1answer
22 views

trying to understant connected component

Now, I am studying connected component. It is known that connected component is an equivalence classes hence every connected components are disjoint. I confuse with term disjoint in here. Unit ...
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22 views

Which the follow property is correct? [on hold]

Let $X$ be a metric space and $E‎\subseteq‎ X$. Which the follow property is correct? for every $E‎\subseteq‎ X$,$\overline{E^\circ}=\overline{E}$ for every $E‎\subseteq‎ ...
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1answer
35 views

Is there exist a ball with lesser radius than another ball that contains it?

If $B_1$ and $B_2$ are two balls in metric space $X$ with radius $r_1$ and $r_2$, respectively and $B_1‎\subseteq‎B_2$,Is it possible that $r_1>r_2$ ? I think, it can occure in discrete metric ...
2
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1answer
24 views

Convergent sequence in box topology

Given $P = \left\{a\in R^R: a_n > 0 \ \forall n\in R\right\}$. Let 0 $= (0_n)_{n\in R}$. Assume $R^R$ has the box topology. Prove that: (a) 0 is in Cl($P$) (i.e, closure of $P$) (b) No sequence ...
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2answers
21 views

Existence of Divisor-Zeros in $c(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
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1answer
18 views

If $A$ , $B$ are dense in the metric space $X$ then,…

Let $X$ is a metric space and $A$ and $B$ are two dense subset in $X$. Which is correct? if $A$ is open, $A‎ \cap‎‎B$ is dense in $X$ if $A$ is closed in $X$, $A‎ \cap‎‎B=\emptyset$ $(A-B)\cup(B-A)$ ...
3
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1answer
38 views

What is the closure of $A$ in $\mathbb{R}$

If the set $A$ has been as follows, find the closure of $A$ in $\mathbb{R}$. $$A=\left\lbrace \frac{m+n}{2m+n+1}: m,n\in\mathbb{N}\right\rbrace$$
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1answer
23 views

Existence of continuous function in metric space

Given a metric space $(X,d)$ with closed set $C$ ($C\neq \emptyset$) and a point $m\in X-C$. Prove that there exists a continuous function $f: X\rightarrow [0,1]$ such that: (a) $f(m) = 0$ and ...
2
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1answer
13 views

When can I paste together two homeomorphisms?

Let $p: X\to Y$ and let $U,V$ be open in $X$. Assume $p$ is a homeomorphism when restricted to $U$ and when restricted to $V$. Under what conditions is $p$ a homeomorphism on $U\cup V$? This came up ...
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13 views

Proving for every subset $A$ of $X$ $Int(Cl(A))$ is a regular open set.

Ok, so I ham working through WIllard's topology (I'm loving this book). Here is problem 3D3 found in page 29: let $X$ be equipped with a topology, we have to prove that for all $A\subset X$ we have ...
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0answers
38 views

a question about functional analysis, and this question is about Riesz's theorem(how to prove it?)

Riesz's theorem: Let $(V,\|\cdot\|)$ be a normed vector space, and suppose $C$ is a compact subset of $V$, moreover, $C$'s interior is not empty, and then please prove $\dim(V)<\infty$. ...
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1answer
18 views

the $\sigma$ algeba generated by the class of open intervals with rational end points coincide with the borel $\sigma$ algebra on the real line.

Show that the $\sigma$ algeba generated by the class of open intervals with rational end points coincide with the borel $\sigma$ algebra on the real line. I tried to solve the question but I cannot ...
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1answer
16 views

Convergent nets in a topological space

Let $X$ be a set and $\mathcal{T}_1$, $\mathcal{T}_2$ be two topologies on $X$. Suppose the following is true: for any net $N=\{x_\alpha\}_{\alpha\in A}$ in $X$, $N$ is convergent in ...
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1answer
17 views

Question about the Baire space, $\sigma$-algebra and $\sigma$-ideal.

Let $\text{BP}(X)$ denote $\sigma$-algebra of subsets of $X$ with the Baire Property BP and $\text{MGR}(X)$ denote the $\sigma$-ideal of meager sets in $X$. Assume $X$ is second countable Baire ...
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1answer
33 views

Finite Complements Topology and Convergent Sequences

Let $X$ be the set of natural numbers $\mathbb{N}$ together with $\mathcal{F}$, the finite complements topology. I've been asked to determine two different sequences $(x_{n}), (y_{n})$ such that: ...
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1answer
37 views

Homeomorphism between lower limit topology and another topology

Given a basis $B$ for a topology $T$ on R with $B=\left\{[a,b): a,b\in R-\left\{0\right\} \cup \left\{(-x,x): x>0\right\}\right\}$. Show that $(R,T)$ is homemorphic to the lower limit topology ...
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25 views

Question about of Baire property and Baire space

In reading Kechris book. Please,I would like help with this proposition. For convencion we put for $A \subseteq X$, $$\sim A=X\setminus A$$ If $A$ is comeager in $U$, we say that $U$ forces $A$, ...
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1answer
25 views

Non-compactness of a set

Let $\mathbb{R}^{\infty}$ be the set of all "infinite-tuples" $x=(x_1,x_2,...)$ of real numbers that end in an infinite string of $0$'s. Define an inner product on $\mathbb{R}^{\infty}$ by the rule ...
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1answer
17 views

Finite Complements Topology [on hold]

Let $X$ be any set and $\mathcal{F}$ the finite complements topology. Prove that if $(X,\mathcal{F})$ is a topological space then $(x_{n}) \rightarrow p \in X \Leftrightarrow \{ n \in \mathbb{N} ...
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1answer
19 views

The maximality of product toplogy w.r.t a statement

Product space, $\prod\limits_{a \in I} {X_a}$, where $I$ is from arbitrary index set. Let $W$ be any topological space, $f: W \to \prod\limits_{a \in I} {X_a}$ is continuous iff $ \pi_b \circ f: W ...
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1answer
31 views

discrete family of sets

Suppose $X$ is a topological space and $\mathcal{F}$ is a discrete family of close subsets of $X$. Then is it true that, any two members of $\mathcal{F}$ are disdoint?? [A family $\mathcal{F}$ is ...
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1answer
14 views

Question about hereditary separability

If we have the clousure of a metric space $\overline{X}$ and we know that $X$ is separable (so in fact it's hereditarily separable), can we say that $\overline{X}$ is hereditarily separable too? ...
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34 views

Prob. 8(c) in Exercises after Sec. 17 in Munkres' TOPOLOGY, 2nd ed: Does equality hold? Or does either inclusion hold?

Let $X$ be a topological space; let $A$ and $B$ be subsets of $X$. Then what is the relation between the closure $\overline{A \setminus B}$ in $X$ of the set $A \setminus B$ and the difference ...
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2answers
22 views

Prob. 5 in Exercises after Sec. 17 in Munkres' TOPOLOGY, 2nd ed.: How to prove this result in a general ordered set?

Here's Problem 5 in the Exercises after Section 17 in the book, Topology by James R. Munkres, 2nd edition: Let $X$ be an ordered set in the order topology. Show that $\overline{(a,b)} = [a,b]$. Under ...
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1answer
23 views

Prove that the distance between a disjoint compact set and closed set is nonzero.

Let $X$ be a metric space, and $K$ is compact and $C$ is closed, and $K$ and $C$ are disjoint. Prove that $$\inf_{k\in K, c\in C} d(k,c) > 0$$ What I'm thinking is considering the function $f: K ...
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ISO Hurewicz Wallman *Dimension Theory*

Is there an electronic copy floating around? I assume it is public domain now.
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1answer
22 views

How do I prove the interior of subspace $\ell^1$?

Let $E:=\ell^1$ is Banach space with standard norm for $\ell^1$, $P:=\{\bar{x}\in\ell^1: \bar{x}=(x_i)=(x_1,x_2,\ldots),x_i \geq 0, \forall i \in \mathbb{N}\}$ and defined that interior of $P$ is ...
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1answer
38 views

Product of infinite discrete space is second countable

Given $K^w$ equipped with product topology is an infinite product of countably infinite discrete space $K$ . Show that $K^w$ is second countable. My Progress: Since the product topology means there ...
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1answer
24 views

Prob. 10 after Sec. 16 in Munkres' TOPOLOGY, 2nd edition: Which of these topologies is finer than which?

Let $I = [0,1]$, the unit closed interval on the real line with its usual order. Compare the product topology on $I \times I$, the dictionary order topology on $I \times I$, and the topology $I \times ...
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1answer
44 views

Set and its complement that are both dense

I'm trying to use Baire's theorem to give an example about open sets $\left\{X_i:i\in N\right\}$ in $\Bbb R$ such that $\cap_{i\in N} X_i$ and $\Bbb R - \cap_{i\in N} X_i$ are dense in $\Bbb R$. So ...
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1answer
35 views

Every closed subset of $\mathbb R^2$ is the frontier of a set?

I would like a proof verification, alternative proofs and in general some backround to this problem or related problems: A closed subset of $\mathbb R^2$ with the euclidean topology is the frontier ...
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17 views

some new Compact Subsets of $M_n(\mathbb C)$ or $M_n(\mathbb R)$

Can we list the compact subsets of $\mathbb R^n$. I found this is very useful for those who preparing for competitive exams. I am giving some thing.. Set of all orthogonal matrices The proof is ...
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1answer
32 views

Using the open cover definition of compactness to show that the set of nilpotent $m \times m$ real matrices is noncompact

Is the set of nilpotent $m \times m$ real matrices compact? I found the proof of this statement, using Heine-Borel theorem on $\mathbb R^n$. Tha'ts quite good. But, is it possible to prove this ...
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36 views

Classify Maps up to Homotopy from Special Cylinder to $S^4$

Let $n\in\{0,1,2,3,4,5\}$ be given. Let $F \subseteq \{1,2,3,4,5\}$ be given such that $|F| = n$. Define an $F$-flip map on $S^4\to S^4$ by sending $x_j \stackrel{F\mbox{-flip}}{\mapsto} ...
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1answer
32 views

Good Pair corollary of the excision theorem

I have problem with understanding the following proof $q_*$ is isomorphism as q is a quotient map and so outside A, it is a homeomorphism implies that $q_*$ induces isomorphism. Given the above ...
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1answer
30 views

Why $d(A,B)$ is not always achieved when $B$ is closed?

We know that in a metric space $(E,d)$ if $A,B\neq\emptyset$ are compact and disjoint then there exists $a\in A$ and $b\in B$ such that $d(a,b)=d(A,B)$ but how to prove that if $A$ is compact and ...
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1answer
35 views

Compactness and open sets

I have this small question, if $(E,\tau)$ is a Hausdorff space and $A,B$ two separated compact sets, how to prove the existence of two open disjoint sets $U$ and $V$ such that $B\subset V$ and ...
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2answers
19 views

Let A be a non empty subset of R .Let I(A) denotes set of interior points of A Then

Let $A$ be a nonempty subset of $\mathbb{R}$ .Let $A^\circ$ denotes set of interior points of $A$ Then $A^\circ$ can be A. Empty B. Singelton C. A finite set containing more than 1 element D. ...
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16 views

an open subspace of locally compact is dense

Let $X$ be locally compact Hausdorff. Then a subspace $A$ of $X$ is dense and locally compact iff $A$ is open. I can prove the necessary condition. But for the sufficient condition, I can not get ...
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1answer
45 views

Hatcher, Thm 2.13

Theorem 2.13. If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $$\cdots \longrightarrow \tilde H_n ...
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22 views

Definition of a topology on a set $X$: may $I$ contain uncountable many elements or is it restricted to be finite or countable?

I've the following definition of a topology $\mathcal I$ on a set $X$: (T1) $U_a \in \mathcal I, a \in I \Rightarrow \cup_{a \in I} U_a \in \mathcal I$ (T2) $U_1, U_2 \in \mathcal I \Rightarrow U_1 ...
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2answers
24 views

an open subspace of compact space

It is know that every compact subspace of Hausdorff space is closed and every closed set is compact. So I have a question as folows: is there any compact non-Hausdorff space $X$ such that every open ...
4
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1answer
66 views

Is really $[0,1]\times[0,1)$ homeomorphic to $[0,1)\times[0,1)$ with the product topology?

I got asked to show that $[0,1]\times[0,1)$ is homeomorphic to $[0,1)\times[0,1)$ with the product topology, but I'm having trouble understanding why that's not false. $[0,1]$ is obviously compact ...
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0answers
19 views

Simple question about of the formulas concerning the forcing relation $U \Vdash A$

If $A$ is comeager in $U$, we say that $U$ forces $A$, in symbols $$U \Vdash A$$ A weak basis for a topological space $X$ is a collection of nonempty open sets such that every nonempty open set ...
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1answer
21 views

Definition of 'saturated' set?

My notes from the lecture says "Let $X$ be a topological space and let $R$ be an equivalence relation. Then, $A \in X$ is called saturated with respect to $R$ if it is a union of equivalence classes." ...