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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24 views

Showing this set $A$ is closed, bounded and not compact?

Let $$ l^1(\mathbb{N}) = \left\{ (x_n)_n \mid \sum_{n = 0}^{\infty} | x_n | \ \text{converges} \right\}, $$ the space of all sequences whose associated series converge absolutely. On this space we ...
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22 views

$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
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17 views

Topological Join of Unit Balls

I have seen that apparently one has for spheres that $S^n*S^m=S^{n+m+1}$. Is there a similar result for unit balls? Thank you.
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2answers
18 views

$A$ be closed in $X$ , $U \subseteq A$ is open in $A$ , $V$ be open in $X$ s.t. $U \subseteq V$ , then is $U \cup (V\setminus A)$ open in $X$?

Let $X$ be a topological space , $A$ be closed in $X$ , $U \subseteq A$ is open in $A$ , $V$ be an open subset of $X$ such that $U \subseteq V$ , then is it true that $U \cup (V\setminus A)$ is open ...
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1answer
17 views

$S$ be $\pi$-system on a set, given two measures on $\sigma(S)$, is there a topology on $\sigma(S)$ making $S$ dense, and the two measures continuous?

Let $\Omega$ be a non-empty set , $S \subseteq \mathcal P(\Omega)$ be a Pi system (https://en.wikipedia.org/wiki/Pi_system ) on $\Omega$ , let $\sigma(S)$ be the $\sigma$-algebra generated by $S$ (i.e....
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21 views

Alternative proof: show that any metrizable space $X$ is normal - Part 2

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero ...
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2answers
53 views

Baby Rudin Problem 2.16

Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q) = |p-q|$. Let $E$ be the set of all $p\in Q $ such that $2<p^2<3$. Show that $E$ is closed and bounded in $Q$, but $E$...
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1answer
74 views

Can a metric space over integers induce a topology?

Questions to get a better grasp of basic topology: A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e., a function $$ d \colon M \times M \to \...
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1answer
33 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
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1answer
36 views

What subbase generates metric topology?

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems ...
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1answer
27 views

$\pi_0(SO(N))$ and $\pi_0(O(N))$: Inconsistency between Bott periodicity and basic understanding of $\pi_0$

I need to know the homotopy groups of the oriented Grassmannian $\widetilde{Gr}(\infty,\infty) \cong \lim_{N \rightarrow \infty} SO(2N)/(SO(N) \times SO(N))$, and I've run into an inconsistency. It ...
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0answers
14 views

Topology induced by a proximity space is always completely regular - proof

https://en.wikipedia.org/wiki/Proximity_space says "The resulting topology is always completely regular. This can be proven by imitating the usual proofs of Urysohn's lemma, using the last property of ...
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36 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
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37 views

Is there any relationship between topological and graphical connectedness?

We have two ideas of contentedness from two different branches of mathematics - Topology and Graph Theory. One talks about the connectedness of a space and another about a graph. But does there exist ...
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1answer
58 views

Is there a topology such that $(\Bbb R, +, \mathcal T)$ is a compact Hausdorff topological group?

I already know that this is impossible for $(\Bbb Q, +, \mathcal T)$ to be a compact Hausdorff topological group (notice that the trivial topology does not work because it is not Hausdorff). Indeed, ...
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2answers
59 views

Categorical Interpretation of Strongest/Weakest Topology

One way to define the product topology is as the weakest topology for which all projection maps are continuous. Strongest/weakest topologies satisfying a given property are ubiquitous in topology and ...
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2answers
24 views

Question on finite subfamilies of an infinite family of sets

Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$. Suppose $\bigcap X\subseteq B$. Is it ...
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2answers
34 views

Analytic curve divides disk into two Jordan regions

Let $\gamma:(0,1)\rightarrow\mathbb{C}$ be an analytic Jordan arc. It seems natural to me that for every $\gamma(t_0)$ we can find a disk with center $\gamma(t_0)$ that is divided by $\gamma$ into two ...
2
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1answer
35 views

Determining Class of a general Borel measure

Let $(X, \mathcal{T})$ be a topological space, and $\Sigma = \Sigma(\mathcal{T})$ the $\sigma$-algebra of Borel sets (that is, the $\sigma$-algebra generated by $\mathcal{T}$). In Real Analysis and ...
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1answer
38 views

Preservation of connected component in enlarged space

This is something I came up with while trying to solve another problem; it's a basic point-set topology problem that seems true and easy to prove, but I keep getting stuck. Let $A,B$ be subspaces ...
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1answer
16 views

Show that a linear continuous functional can be expressed as finite linear combination of a collection of functionals separating points.

Let $\{l_\alpha\}$ be a collection of linear functions in a linear space $X$ over $\mathbb{R}$ that separates points. Put $\tau$ as the weakest topology in which all $l_\alpha$ are continuous (Such ...
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0answers
15 views

Product and Join of $G$-CW-Complexes

Given a topological group $G$ and two $G$-CW-Complexes $X$ and $Y$ I want to understand the natural CW-structure on $X\times Y$ and $X*Y$. I understand that the concepts are very similar, so I want to ...
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1answer
17 views

Proving $\mathbb{R}^2$ is not separable for this metric?

Let $d_S$ be a metric on $\mathbb{R}^2$ defined as follows $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{...
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27 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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0answers
19 views

How to got there are only two kinds of 1-dim manifold 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 ?
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10 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
45 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
23 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
29 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
19 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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32 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
38 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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0answers
38 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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64 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
33 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
26 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
24 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\...
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1answer
31 views

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

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 ...
3
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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
20 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
49 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
30 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
15 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 $x_n$ ...
1
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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{...
2
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1answer
38 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 ...