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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1answer
28 views

How many closed surfaces (up to homeomorphism) are there with Euler characteristic -2? [on hold]

I was thinking of splitting up the cases of orientable and non-orientable surfaces.
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15 views

Set interpretation - topology vs probability

Consider the sequence of i.i.d. distributed random variables $(X_i)_{i\geq1}$ on $\mathbb{Z}^d$. We define the following norm $I(x)=\mid x\Gamma^{-1}x\mid$, where $\Gamma$ denote the covariance matrix ...
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2answers
21 views

Error in proof that the closure of open ball equal the closed ball in all metric spaces

Let $(X, d)$ be a metric space. Denote the open and closed ball as $$B(x_0, r) = \{x \in X \mid d(x, x_0) \lt r\},$$ $$D(x_0, r) = \{x \in X \mid d(x, x_0) \leq r\}.$$ Then $\overline{B(x_0, ...
-1
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0answers
31 views

Question about n- expansive homeomorphism

Let $(X, d)$ be a compct metic space and $f$ be a homeomorphism on $X$ . Suppose $\Gamma_c(x)=\{y: d(f^{n}(x), f^{n}(y))<c \ , \forall n\in Z\}$ and for some $z\neq x$, $z\in \Gamma_c(x)$. ...
3
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1answer
46 views

Torus with a point deleted is not a retract of the torus.

Show that the 2-torus with a deleted point $T\setminus \{ x_0\}$ is not a retract of $T$. I know that we can prove the torus with a point removed deformation retracts to the wedge of two circles. ...
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0answers
24 views

find the interior and closure of a set under topologies [on hold]

What is the interior, boundary the set $$\{1/n \mid n\in \mathbb{N}\}$$ under the standard topology? What about: trivial topology and upper limit topology? Many thanks
3
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0answers
23 views

Compact-open topology on $\operatorname{Hom}_\mathbb{R}(V,W)$

Suppose $V$ and $W$ are finite-dimensional real vector spaces, and I give $\operatorname{Hom}_{\mathbb{R}}(V,W) \cong V^* \otimes W$ its usual vector space topology. Does this agree with the subspace ...
2
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1answer
34 views

Show that finite dimensional subspace is closed

We know that if $V$ is a normed vector space and $W$ is a finite dimensional subspace of $V$, then $W$ is closed. One way to prove this is to show that $W$ is actually complete. Since complete space ...
2
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0answers
27 views

Question about a map of spheres and it's fundamental group.

Let $j:S^1\longrightarrow S^4$ be an embedding. Can $\pi _1(S^4\setminus j(S^1))$ be trivial? If it is not trivial, can the fundamental group $\pi _1(S^4\setminus j(S^1))$ be abelian? This is what ...
2
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1answer
27 views

Prove that $l_2$ is a second countable space and is separable.

This is actually a problem from Schaum's Outline Series: General Topology. Let $\{u_n\}$ be a sequence of real numbers such that the series $$\sum_{n=1}^\infty u_n^2<\infty.$$ That is, the series ...
4
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0answers
52 views

Compact sets of compact-open topology

Let $X$ and $Y$ be topological spaces,$X$ not compact and $Y$ metric, denote with $C(X,Y)$ the set of continuous functions between $X$ and $Y$ and put on $C(X,Y)$ the compact-open topology. My ...
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0answers
41 views

Proof verification: Munkres Theorem 22.1

In Step 2 of Theorem 22.1 from Munkres' Topology: Let $p:X\to Y$ be a quotient map; let $A$ be a subspace of $X$ that is saturated with respect to $p$; let $q:A\to p(A)$ be the map obtained by ...
1
vote
1answer
54 views

Is the Euclidean=usual=standard topology on $\mathbb{R}^n$ kind of like the discrete topology?

It's more of a conceptual question. The discrete topology, I understand, is essentially the power set, so every possible subset put together for a set $X$. But when it comes to $\mathbb{R}$...the ...
2
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2answers
27 views

Example of a Hausdorff Space with a point that has different character relative to a dense subspace

I am looking for a Hausdorff space $X$ that contains a dense subset $D$ and a point $x \in D$ such that $\chi(x, D)<\chi(x, X)$. I know that $\leq$ always holds, and I also know that $X$ cannot be ...
0
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2answers
22 views

Topology induced by metric and subspace topology

Let $(X, d)$ and $(Y, d_Y)$ two metric spaces, where $Y \subset X$ and $d_Y=d_{|Y}$. On $Y \subset X$ there are two topologies, which are the subspace topology $T_{d}^Y$ induced by topology $T_d$ on ...
0
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2answers
37 views

Not very sure about this contraposition…open and closed sets

I have this lemma that states, Let $X$ be a topological space and $A \subseteq X$. Then, $A$ is open in $X$ if and only if $\forall x \in A$, there is a neighborhood of $x$ that is contained in ...
0
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1answer
14 views

Regularity connected union of smooth sets

Consider a countable family of bounded, open connected sets with smooth boundary $S_i \subset\mathbb{R}^n$. What can be said about the regularity of the boundary of $S = \bigcup_{i = 1}^{\infty} S_i$ ...
1
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2answers
25 views

How to show that countable union of $F_\sigma$ is $F_\sigma$

On https://www.physicsforums.com/threads/countable-intersection-of-f-sigma-sets.666055/ Is it claimed that it is obvious that countable union of $F_\sigma$ is $F_\sigma$ Can someone elaborate why ...
0
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2answers
43 views

How to prove any closed set in $\mathbb{R}$ is $G_\delta$

Is there a general approach to show that any closed set in $\mathbb{R}$ is $G_\delta$? To show any open set is $F_\sigma$, the approach is: Show all open intervals are $F_\sigma$ Show all open sets ...
1
vote
1answer
37 views

$\operatorname{S}X \approx I\times X/ \{0\}\times X\cup I\times \{x_0\}\cup \{1\}\times X$

The reduced suspension of the pointed space $(X,x_0)$ is the smash product $(\mathbb S^1\wedge I, *)$ of $(X,x_0)$ with the $(\mathbb S^1,s_0)$ and is denoted by $\operatorname{S}X$. My problem is to ...
0
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0answers
26 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
1
vote
1answer
33 views

Is there a bijection from 3-dimensional to 2-dimensional cartesian space?

Given a set $ M $ of coordinates in 3-dimensional cartesian space. Is it possible to find a bijection to 2-dimensional cartesian space? (This question arose from a rather practical problem of ...
4
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1answer
296 views

Is it possible to construct Hausdorff compact topology on every set?

I'd like to know if it's possible to construct Hausdorff compact topology on every set. Assume the axiom of choice if needed. Thanks for ideas.
1
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1answer
43 views

Is there a Mobius (infinite) cylinder?

In order to understand the question of the title I need to understand another thing first. If we consider the Mobius band, locally, for a $U_i \subset S^1$, where $S^1$ is the base space, the bundle ...
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0answers
19 views

$star(v)$ for a 0 simplex

Suppose you have a simplicial complex and a vertex $v$ which is not connected to any other vertex. Is $st(v)$ just the empty set? If you're looking at the inside of a simplex you don't look at ...
0
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1answer
26 views

Exhaustion by compact sets in $\mathbb{C}^n$

Let $U\subseteq\mathbb{C}^n$ be open. For every $j\in\mathbb{N}$ define $$K_j:=\{z\in U:\left\|{z}\right\|_{\infty}\le j,d_{\infty}(z,\mathbb{C}^n\setminus U)\ge 1/j\}.$$ Then the following ...
0
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0answers
46 views

Closed set minus an open set.

If $A= [0,1]\cup[2,3]\cup \{4\}$ and $B= (0,1)\cup(2,3)$ What would $A\setminus B$ be? I believe the answer would be $\{4\}$ but I think I may be wrong....
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0answers
26 views

topology notation $G \backslash X$

Suppose there is a group $G$ acting on space $X$. What does the following notation mean? $$G \backslash X$$ Thank you!
1
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1answer
56 views

Continuous function injective over a compact set, and locally injective on each point of the set

Suppose we have a function $F: \mathbb R^n \rightarrow \mathbb R^k$ continuous over some open set $U \in \mathbb R^n$, and let compact set $K \subset U$. $F$ satisfies the following properties: 1) F ...
0
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1answer
12 views

Given that $X$ is a topological space and $X$ is Hausdorff, with $K_1$ and $K_2$ in $X$ being compact sets, show that their intersection is compact.

This question requires me to prove that if $K_1$ and $K_2$ are two compact subsets of $X$, and $X$ is Hausdorff, then their intersection is compact in $X$. I know that in order to show that their ...
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0answers
17 views

Proof of Banach-Tarski? [on hold]

Could you please make me get Banach-Tarski Paradox with mathematical way ? Is there anything that I should know before get this proof ? Because I'm undergraduate and sometimes everything can be tough. ...
0
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1answer
18 views

Relation betwen the base of a product topology and the bases of the topological spaces

I'm taking an introductory course on topology and we have just defined the product topology as follows: Let $(X,T_{X})(Y,T_{Y})$ be topological spaces, then the product topology of $X$ and $Y$ is the ...
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2answers
31 views

Show that Set in $M:=\{x\in \Bbb R^3 : x_1^2\ge2(x_2^3+x_3^3) \}$ is closed

I have to show this regarding the Euclidean metric. I've already shown that it isn't bounded by showing that the $d(x,y)\:\forall x,y \in M$ isn't bounded. I know that in order to show the ...
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0answers
23 views

To prove the properties of Denjoy's Maps

We need to show that the Denjoy homeomorphism constructed may actually be made $C_1$. a)For each integer $n$,let $$l_n=\frac{1}{(|n|+1)((|n|+2)}.$$Show that $$\sum_{n=-\infty}^{\infty} l_n ...
2
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1answer
34 views

Is the space of subsets of $\mathbb R^n$ with the Hausdorff metric separable?

Let M be the Metric Space whose "points" are the Closed and Bounded subsets of a finite dimensional Euclidean Space and whose "distance function" is the Metric defined by Hausdorff for such point ...
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0answers
38 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
3
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1answer
34 views

Fundamental Group and DeRahm Cohomology from Group of Covering Transformations

Old qual problem here, test tomorrow in topology and we barely got to DeRahm Cohomology so I'm not sure how to do this. Let $G$ be the group of transformations of $\mathbb{R}^3$ generated by ...
0
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0answers
25 views

If $X$ is a topological space then $X \times X$ is not homeomorphic to $S^1$. [duplicate]

If $X$ is a topological space then $X \times X$ is not homeomorphic to $S^1$. How can I show this? I know that $\pi_1(S^1)=\mathbb{Z}$ and $\pi_1(X \times X)=\pi_1(X)\times \pi_1(X)$, but I got stuck ...
0
votes
4answers
32 views

Problems understanding “connectedness”

I'm starting a foray into complex analysis, and I've come across the term "connected." I've vaguely heard the term before, but the book ("Complex Variables and Applications, 9th edition" by Brown and ...
0
votes
1answer
20 views

Give an example of a function that is starlit but contains a point which is not a star centre

Having trouble with this one ! I understand that it is starlit if we can draw a straight line segment to every point of the region from the star centre Intuitively I thought that the punctured open ...
0
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2answers
31 views

Characterize those functions which maps open interval to open interval(s).

This question was asked by one of my Professor during the class.I'm getting intuition that these functions should be one-one(I'm wrong maybe).But, i'm unable to classify all such functions. Please ...
2
votes
1answer
33 views

Confusion with Closures in the topological sense

The rigorous definition is A closure of $A \subseteq X$ of a topological space $X$ is denoted Cl($A$) and is the intersection of all closed subsets of $X$ that contain $A$. The more intuitive or ...
2
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2answers
33 views

Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
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0answers
27 views

Does there exist a local homeomorphism between these two set?

We have to decide if there is a local homeomorphism between $\mathbb{R}^{4}$ and (X$\times$Z), where X=$\mathbb{C}^{2}$ with metric topology and Z=$\mathbb{C}^{2}$ with cofinite topology. Here is my ...
1
vote
1answer
47 views

Property of compact metric space

Let $X$ be a compact metric space. Let $A$ be a closed subset of $X$ and let $x\in X$ be a point not in $A$. Show that there exists two disjoint open sets $U$ and $V$ such that one contains $A$ and ...
0
votes
1answer
22 views

Degree 3 map from the torus to the sphere.

Construct a degree 3 map from $T$ to $S^2$ where $T$ is the torus? I can find a degree 0 and degree 1 map by the following proof: Embed $T$ in $\mathbb{R}^3$ in the usual way. Consider any point ...
1
vote
1answer
33 views

Composition of homeomorphisms $[0,1] \to [0,1]$

Let $f:[0,1] \to [0,1]$ be a homeomorphism with $f(0)=0$ and $f(1)=1$. If $f$ is not the identity map, is it true that $f^n \neq f$ for all integers $n>1$? Edit: By $f^n$ I mean $f$ iterated $n$ ...
0
votes
2answers
44 views

$(X \times X) /{\sim'}\cong (X/{\sim}) \times (X/{\sim})$

The full description of this problem is: Let $X$ be a topological space. Let $\sim'$ be the equivalence relation on $X\times X$ defined by $(x,y)\sim'(x',y')$ iff $x \sim x'$ and $y \sim y'$ ...
2
votes
2answers
26 views

Covering map between two path connected sets

First off, I see a lot of variations of this problem cropping up on practice qualifiers, and I'm trying to regain my knowledge of topology. Let $p: X \to Y$ be a covering map where $X$ and $Y$ are ...
1
vote
1answer
38 views

Infinitely countable subset of $\mathbb{R}^2$ is connected.

Let $A\subset \mathbb{R}^2$ be an infinite countable subspace. Can $A$ be connected? I have seen plenty of proofs that show that $\mathbb{R}^2\setminus A$ is connected and I understand those. ...