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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Looking for a general point that can be found in the highest dimensional irreducible component of the $k$-secant variety.

Let $X\subseteq\mathbb{P}^{n}$ be an irreducible variety, $p_{1},\ldots,p_{k}\in X$ general points and $p\in\langle p_{1},\ldots,p_{k}\rangle$ a general point. I want to show that we can consider $p$ ...
2
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0answers
21 views

About contractible implies simply connected

most books say that contractible implies simply connected is trivial as at time one, every point of the space is shrunk to one point, including the loop. However, there is a problem. In discussing ...
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1answer
37 views

Non-continous topology?

I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I ...
2
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1answer
13 views

Decomposition into irreducibles of a Noetherian topological space

I'm struggling with the proof that says a Noetherian topological space $X$ is the finite union of closed irreducible subsets. In particular with this part: First observe that every nonempty set of ...
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1answer
25 views

if two space are homotopy equivalent and one is connected, prove that the other is connected as well

I've tried using the definition of homotopy equivalent spaces which states that X and Y are homotopy equivalent if: There are continuous functions $f:X \rightarrow Y,g:Y \rightarrow X$ such that $f ...
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7 views

Are there any other measures of complexity for a continuous map than topological entropy?

Let $X$ be a compact topological space and $T\colon X\to X$ be continuous. In order to say something about the complexity of $(X,T)$ there is of course the notion of topological entropy of $T$, ...
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7 views

Forms on Varieties

Given M n-dimensional differentiable variety, orientable and compact, and $w_0$ an orientation form of M, we want to proof: i) Given $\rho \in \Omega^r(M)$, $0 \leq r \leq n$ and $g: W \rightarrow M$ ...
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1answer
23 views

Categorical version of the Tietze Extension Theorem

In Donald Hartig's short paper An Important Functor in Analysis and Topology, Theorem 1 is preceded by the following statement: Since the spaces we are dealing with are compact, a one-to-one map ...
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2answers
22 views

Existence of a countable basis in the definition of a manifold and uncountable bases.

In the definition of a manifold $M$ of dimension $n$ in An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby (page 6), the third criterion is $M$ has a ...
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4answers
47 views

How does this specific set look like?

How does the set $ \begin{pmatrix} \cos(x) \\ 4\sin(x) \\ \end{pmatrix} \in \mathbb R ^2$ with $x \in \mathbb R$ look like? I guess it should be similar to $f(x) = ...
2
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1answer
20 views

Characterize R0-space by convergent filters

I want to prove the equivalence of the two following characterizations of R0-spaces. One comes from my textbook (with filters), the other one is taken from wikipedia. First, I will introduce the ...
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1answer
52 views

Characterization of 1-dimensional manifolds. [duplicate]

My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$. Is this true? If yes, is it possible to prove it from first ...
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14 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
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1answer
19 views

Why boundary of a locally closed set is nowhere dense?

Let $X$ is locally closed , i.e. exist open $U$ S.t. $X=\overline{X} \cap U $ , and $bd (X) = \overline{X} \setminus \mathring{X} $. How can I show that $ bd(X) $ is nowhere dense? I read topics ...
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5answers
88 views

Why is $\operatorname{Int}(A) \cup \operatorname{Int}(B) \neq \operatorname{Int}(A \cup B)$?

I know that $\operatorname{Int}(A) \cup \operatorname{Int}(B) \subset \operatorname{Int}(A \cup B)$, but that the other direction does not hold, so can anybody please tell me whats wrong with the ...
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3answers
29 views

How continuity of $f$ and path-connectedness of $g$ results in $f\circ g$ to be path-connected?

Theorem 6.29 (p.213) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa says: Assume that $f : X \rightarrow Y$ is continuous and $X$ is path connected. Then $f (X)$ is a ...
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0answers
32 views

Maps between groups and classifying spaces

Suppose we have two Lie groups $G$ and $H$, as well as two homomorphisms $\phi_1,\phi_2 \colon G \to H$ and an arbitary continuous map $g \colon G \to G$. Futhermore suppose that $\phi_2$ is homotopic ...
5
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1answer
43 views

Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Once I heard that there exists two compact spaces $K_1$ and $K_2$ which are non-homeomorphic, but with $K_1\oplus K_1$ homeomorphic to $K_2\oplus K_2$ (where $\oplus$ denotes the topological sum). Is ...
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2answers
54 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
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1answer
48 views

Simples curves on $RP^2$

A subset $\Sigma $ of a space is a simple closed curve if it is homeomorphic to S1. Let $p: S^2 \rightarrow RP^2$ be the canonical projection of the sphere onto the projective plane. Prove that if ...
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1answer
42 views

Are these two definitions of basis equivalent?

Lecture note definition Let $(X, \mathcal{T})$ be topological space, A $basis$ of $\mathcal{T}$ is a collection $\mathcal{B}$ of open sets satisfying the following: For each open set $U$ and ...
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2answers
32 views

$X,Y$ are compact Hausdorff. $f$ is bijective continuous. Is $f$ a homeomorphism?

Let $X,Y$ be compact Hausdorff spaces. Let $f:X\to Y$ be one-to-one, onto and continuous. Show that $f$ is a homeomorphism. I came up with this "proof" but I am very sure it is wrong. In order to ...
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2answers
40 views

construct an example of not dense sets [on hold]

Construct an example of a topological space $X$ and $A \subset X$ where $\operatorname{int}(A) \cup \operatorname{int}(X \setminus A)$ is not dense in $X$ which example?
2
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1answer
13 views

Every locally finite family of non-empty subsets of a Lindelöf space is countable.

I just don't understand the conclusion of the lemma: $|\mathcal{A}| \le \aleph_0$. I think it's related with the fact that every member of $\mathcal{U}$ meets only finitely many members of ...
3
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1answer
68 views

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$ if $m\ne n$. You may assume that $S^m$ and $S^n$ are different homotopy type if $m\ne n$. My attempt: Suppose $\mathbb{R}^m$ is ...
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0answers
25 views

Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
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1answer
23 views

Closed and boundary subsets

Let $X$ be a nonempty compact space and let $F_1, F_2, ...$be its closed and boundary subsets. Prove that $\bigcup_{n=1}^{\infty} F_n \neq X$ I have no idea how to do it. My only plan would be to ...
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1answer
15 views

Paracompact topological space: Why is $\overline{V}_s=F_s$?

Question: Why do they say in the remark that $\overline{V}_s=F_s$? Attempt: The only explanation I can think of is that the union $V_s=\cup_{s(t)=t} A_t$ is finite, and I tried to prove it using ...
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1answer
43 views

Countable and not closed subset of infinite compact space

The taks is: Show that in every infinite compact space there is a countable subset that is not closed. At first I read that it should be closed and I had an idea to take a point $x_1 \in X$ and an ...
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1answer
21 views

The Intersection of Equivalence Relations which cover a relation

Exercise A.3 From John Lee( Topological Manifolds) Let $R \subset X \times X$ be any relation on $X$, and define ~ to be the intersecction of all equivalence relations in $X \times X$ that contain ...
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1answer
22 views

interior, exterior and boundary

Prove $b(int(A)) \subset b(A) $ where $b$ is boundary, $int$ is interior and $ext$ is exterior if $x \notin b(A)$ then $ x \in int(A) \cup ext(A) $ if $x \in int(A) \to x \in int(int(A))$ ...
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0answers
26 views

Connected set in normed space

I have this exercise: "let $E$ be a normed space and $X\subset E$ $$X~\text{connected}~\Longleftrightarrow \forall A\subset X,~\text{such that} A\neq\emptyset, A\neq X~\text{we have}~ Fr(A)\neq ...
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1answer
42 views

Non-Empty Finite Subset $U$ of $\mathbb{R}$ is not Open

Consider $(\mathbb{R}, \mathcal{T})$ standard topology Definition : $ U \in \mathcal{T}$ if $\forall x \in U, \exists \delta$ such that $(x-\delta,x+\delta) \subset U$ If using this definition, ...
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0answers
18 views

If $D$ from $X*X $ to $R$ with this condition that $ D(x,y)=-D(y,x)$, and if $ D(x,y)\ge0$, $D(y,z)\ge0$, can we implies that $D(x,z)\ge0$? [on hold]

If there is a function $D$ from $X*X $ to $R$ with this condition that $ D(x,y)=-D(y,x)$, and if $ D(x,y)\ge 0$, $D(y,z)\ge0$, can we implies that $D(x,z)\ge 0$?
3
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0answers
100 views

What do Set-Theoretic (General) Topologists study? [on hold]

I was reading in Elementary Topology by O Viro, O Ivanov, V Kharlamov, and N Netsvetaev and it caught my attention the following quotes by the authors: "...As a research field (refering to General ...
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0answers
52 views

A function continuous on rational points and discontinuous on irrational points [duplicate]

How to find function $f : \Bbb R \to \Bbb R$ such that $f$ is continuous on the rational numbers and discontinuous at irrational numbers? I was told to use the Baire Theorem to show that the set of ...
2
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1answer
45 views

Topologies on a finite set. An open problem?

Some time ago an eminent professor told me about an OPEN problem: Number of possible topologies on a finite set? I was excited about the idea of solving this problem but could not. This was more ...
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2answers
27 views

Interior, closure, isolated points and boundary of a set of a normed vector space

Let $X =(\mathbb{R}^2,||(x_1,x_2)|| := |x_1| +|x_2|)$ be a normed vector space. Find the interior, closure,Isolated points, and boundary of $Y =\{(x, \frac{1}{n})~|~ x\in \mathbb{R} \wedge n\in ...
2
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2answers
38 views

Meaning of n-connected pairs

A topological space $X$ is $n$-connected if the homotopy groups $\pi_r(X)$ for $0 \leq r \leq n$ are trivial groups. This means (let's say geometrically), $X$ is $0$-connected if it is non-empty and ...
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0answers
25 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
2
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2answers
51 views

minimal embeddings of topological spaces into connected spaces

Defintions: Let $X$ be a topological space. 1) A connected space $Y$ is a minimal connected ambient (m.c.a for short) space for $X$ if there exists an embedding $i:X\mapsto Y$, and for every ...
3
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1answer
25 views

Filter of sets containing a subset converges

I'm just learning about filters, and I came across the following exercise in Willard's Topology: Let $X$ be a topological space and $A \subset X$. The cluster points of the filter $\mathcal{F} ...
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1answer
25 views

Orthogonal group acts on vector field

I recently had an exam, yesterday acctually, and there was a question that stumped me. The orthogonal group $O(n)$ acts on $\mathbb{R}^n$ by matrix multiplication, show that the orbit space is ...
3
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1answer
44 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
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1answer
31 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
4
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1answer
66 views

Does the product functor preserve quotient maps?

In Hatcher's Algebraic Topology, he presents a proof that if $(X,A)$ satisfies the homotopy extension property, and $A$ is contractible, then $X \simeq X/A$. Part of Hatcher's proof goes: Suppose ...
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2answers
31 views

Is the intersection of two locally compact locally compact?

Taking locally compact as such that every point has a local base of compact neighborhoods, is the intersection of two locally compact subspaces locally compact?
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1answer
41 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
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2answers
46 views

Closed set, closure of a set

Prove if A is open then $A \cap \bar{B} \subset \overline{A \cap B}$ $ A \cap \bar{B}= A \cap (B \cup B')=(A \cap B) \cup (A \cap B')$ $A \cap B \subset \overline{A \cap B} $ then I have to ...
1
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2answers
24 views

a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$

Prove that a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$. The direction $\Rightarrow$ was easy. But I don't ...