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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28 views

Explicit construction of an $\epsilon$ net covering

Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$ X = \bigcup_{i=1}^n U(x_i, \epsilon) $$ where $U$ is the Open Ball centered at $x_i$ ...
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1answer
75 views

How to prove that there are only two kinds of 1-dim manifolds without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
4
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1answer
46 views

Definition of second topological $K$-group of a Banach algebra

The question is a about the definition of the second topological $K$-group of a Banach algebra $A$. I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\...
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0answers
48 views

Weak convergence implies boundedness in norm vector space

Let $(x_n)_{n\geq 1}$ be a sequence in the normed vector space $(X,\Vert\cdot\Vert)$ and let $x\in X$. Show that the following are equivalent. $x_n$ converges weakly to $x$ The sequence $(\Vert x_n \...
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2answers
25 views

Is the following metric topological equivalent to Euclidean metric?

Let $d_S$ be a metric on $\mathbb{R}^p$ defined as $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{when}\ ...
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1answer
32 views

Mapping finite discrete numbers to the infinite set

This is an extension of my earlier question: Mapping discrete numbers Given that we can "map" $\mathbb{N}$ to $\mathbb{Z}$ via a bijection, I then wondered if it is possible to map a small subset of $\...
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2answers
21 views

Mapping discrete numbers

I would like to find a way to map the natural numbers, $\mathbb{N}$, to integers, $\mathbb{Z}$, and vice-versa. An analogous solution for continuous numbers would be using the $\log()$ and $\exp()$ ...
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2answers
33 views

Show that the only intervals having the fixed point property are the closed intervals.

By Fixed Point Theorem, I know that it deals with closed interval, for eg, [0,1]. And this theorem will be false if [0,1] is replaced by (0,1). The counter example will be $f:(0,1)\rightarrow (0,1)$ ...
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2answers
22 views

discrete space $(X, \tau)$ and indiscrete space $(X,\tau')$ with $X$ has at least two points do not have fixed point property

Let $X$ be a set with at least two elements. Prove that the discrete space $(X,\tau)$ and the indiscrete space $(X,\tau')$ do not have the fixed point property. For the indiscrete space, I think ...
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0answers
39 views

a space isomorphic to $S^{p+q}$

In one of the paper I have met that $$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \mathbb{R}^q \cup \mathbb{S}^{p-1}$$ I don't know how to deal with $\mathbb{S}^{p-1}$ sphere. Are there any ...
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39 views

The Fundamental Semigroups of a Topological Object?

I had the following idea for a generalization of the "fundamental group" of a manifold. So the idea basically was that we can consider a manifold $O$ which has boundary $\partial O$, and instead of ...
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68 views

Topology: What does sets in $\bigcup_{i \in \mathbb{N}} A_i$ look like?

Let $\tau$ be the topology on some set $Y$, and $f_i: X \to Y$ be some continuous function. Let $A_i = \{f^{-1}_i(U)| U \in \tau\}$ and $A = \bigcup\limits_{i \in \mathbb{N}} A_i$ My question is ...
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1answer
34 views

Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
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1answer
27 views

Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
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1answer
26 views

About normal spaces and proximities

I am trying to write another proof (using my theory) of Urysohn lemma. This question has appeared during this research. Let $\mu$ be a $T_4$ (normal) topology on some set $\mho$. Let $\delta$ be ...
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0answers
33 views

Order topology on $\omega \cup \{ \omega \}$ is not metrizible

This is an early exersize in Topology and Geometry by Bredon. I don't understand, what is wrong with the embedding $\{ -\frac{1}{n} \mid n \in \mathbb{N} \} \cup \{0 \}$ in $\mathbb{R}$?
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1answer
29 views

Let $f:S^1 \to X$ a continuous function $X$ a topological space. Then $f$ is homotopic to a constant iff $f$ extends to $D$.

Let $X$ a topological space, $D$ a open unitary disc on $\mathbb{R}^2$ and $S^1 = \partial D.$ How to show that $f: S^1 \to X$ continuous is homotopic to a constant map iff there is a continuous ...
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2answers
23 views

A relation between interior and closed sets

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
4
votes
1answer
31 views

Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$? [closed]

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two ...
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1answer
27 views

Lebesgue-Stieltjes measure corresponding to a right continuous increasing function, $m(\{x\}) = \alpha(x) - \alpha(x-)$ for each $x$

Let $m$ be Lebesgue-Stieltjes measure corresponding to a right continuous increasing function $\alpha$. How do I see that for each $x$, we have$$m(\{x\}) = \alpha(x) - \alpha(x-)?$$
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1answer
21 views

Orthogonal lines on Mercator projection?

I am currently struggling with the following task: We have two pairs of latitude/longitude which determine a small line segment It is needed to get two pairs of latitude/longitude for a small line ...
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2answers
52 views

How do I prove that for any finite subsets A and B exists one set R?

How do I prove that for any finite subsets A and B exists one set R $\left | A\cup B \right |=\left | A \right |+\left | B \right | -\left | A\cap B \right |$ Deduce from this an adequate formula ...
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1answer
38 views

bounded components of the intersection of two planar domains

It seems to be intuitively clear that if U is a domain in the plane having a bounded complementary component C, then C is also a complementary component of the intersection of U with an open disk D ...
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0answers
18 views

Over sequential spaces and $B(H)$

We say that a topological space $X$ is sequential if the following holds : If $U$ is sequentially open then $U$ is open. By sequentially open we mean that $x \in U$ and $x_n \to x$ implies that $x_n$ ...
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1answer
39 views

Can we lift paths of a Lie group quotient $G\to G/H$?

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{...
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1answer
40 views

Open sets and annihilator of functions

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
3
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0answers
34 views

Showing that $\sigma$-algebra is uncountable [duplicate]

Suppose $\mathcal{A}$ is a $\sigma$-algebra with the property that whenever $A \in \mathcal{A}$ is nonempty, there exist $B$, $C \in \mathcal{A}$ with $B \cap C = \emptyset$, $B \cup C = A$, and ...
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0answers
25 views

Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? [duplicate]

Suppose $\mathcal{A}_1 \subset \mathcal{A}_2 \subset \ldots$ are $\sigma$-algebras consisting of subsets of a set $X$. Is $\bigcup_{i = 1}^\infty \mathcal{A}_i$ necessarily a $\sigma$-algebra? If not, ...
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0answers
27 views

If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
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0answers
15 views

boundary of starred compact set in the plane.

Is the boundary of a starred compact C set in the $\mathbb R^2$ connected? Let also suppose it to be the closure of an open set.
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1answer
32 views

Indicator function for limsup, liminf [duplicate]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j} A_i.$$Given a set $D$ define the ...
2
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1answer
30 views

Sequences of sets, liminf and limsup [closed]

If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j}^\infty A_i.$$How do I see that$$\...
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3answers
34 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
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2answers
56 views

Continuous functions and metric topology [closed]

Let $X = C[0, 1]$ be the set of all continuous functions $f : [0, 1] \to \Bbb{R}$ (where the domain and codomain have their usual topologies). Let $d_1 : X \times X \to \Bbb{R}$ be the metric on $X$ ...
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1answer
32 views

verifying homeomorphism of orbit space and suggestions for further study

Define an action of $\mathbb{Z}_2$ on $S^1$ by $(0,z)\mapsto z$ and $(1,z)\mapsto \bar{z}$. An orbit of $z$ is then the set $\{z,\bar{z}\}$. I claim the orbit space $S^1/\mathbb{Z}_2$ is homemorphic ...
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52 views

Can the intersection of infinitely many open sets be neither open nor closed?

We all know the example of infintely-many open sets intersecting in a singleton (closed), but is there an example where the intersection is neither open nor closed?
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1answer
30 views

Is it true that $\delta$ is a Lebesgue number for the given cover?

Let $(X,d)$ be a compact metric space. Suppose an open cover of $X$ is given. Let $\{U_i\}_i$ be an its finite subcover. Consider the following function $f:X\to \mathbb{R}$ defined by $x\mapsto \max\{...
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1answer
22 views

In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
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1answer
15 views

Closure of a subspace of a metric space

Say we have $A \subset X$, given $(X, d)$ a metric space. I want to prove that the closure of $A$, that is $$\overline{A} = \{x \in X \ | \ \forall \varepsilon > 0, \ B(x, \varepsilon) \cap A \...
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1answer
60 views

An exact sequence of compact topological groups.

Let $A, B, C $ be abelian topological groups such that we have the following exact sequence : $$0\to A \to B \to C \to 0. $$ Assume also that A, C are compact and all the maps are open. Then it's it ...
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0answers
13 views

Proving some specific space to be normal [duplicate]

Let $f:X\rightarrow Y$ be a closed continuous surjection. Assume that $X$ is normal. Prove that $Y$ is normal. $X$ is normal,then every one-point set in $X$ is closed,so is one-point set in $Y$ ...
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2answers
34 views

Alternative characterization of complete metric space

Let $(X,d)$ be a metric space. It is complete if every Cauchy sequence for $d$ on $X$ is convergent. I've heard an alternative definition of completeness for $(X,d)$: it is complete iff the ...
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1answer
44 views

Continuous maps satisfying a natural stronger condition

Denote the topological closure operator by $\operatorname{cl}(\ )$ (and the interior operator by $\operatorname{int}(\ )$). A map $f:X \to Y$ between topological spaces is continuous iff $\...
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1answer
16 views

Weak and Weak$^{\star}$ topologies: Annihilator

Exercise: Let $E$ be a Banach space. Let $M\subset E$ be a linear subspace and let $f_0\in E^{\star}$. Prove that there exists some $g_0\in M^{\perp}$ s.t. \begin{equation}\inf_{g\in M^{\perp}}\Vert ...
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41 views

Question on closure of intersection os sets

I know that in general $cl(A \cap B) \subset cl(A) \cap cl(B)$. When is it true that $cl(A\cap B) = cl(A)\cap cl(B)$?
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1answer
44 views

Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
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2answers
19 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
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1answer
22 views

Showing $\overline{N(A,\epsilon)}=\{x\in\mathbb R^n \ | \ \rho(x,A)\leq\epsilon\}$

For nonempty $A\subset\mathbb R^n$and a point $x\in\mathbb R^n$, the distance is defined as below. $$\rho(x,A)=\inf\{\|x-y\| \ | \ y\in A\}$$ And in this question, a cover contains $A$ is defined as ...
3
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3answers
29 views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable. We want to construct an infinite sequence of nonempty disjoint measurable sets. ...
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2answers
47 views

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$?

Which of the following subsets of $\Bbb R^2$ are homeomorphic to the set $\{(x, y) \in \Bbb R^2 \mid xy = 1\}$? $a. \{(x, y) ∈ \Bbb R^2 \mid xy − 2x − y + 2 = 0\}.$ $b. \{(x, y) ∈ \Bbb R^2 \...