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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2
votes
2answers
54 views

proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
0
votes
1answer
46 views

can open and not open sets in $\mathbb{R}^n$ be homeomorphic?

Can an open set in $\mathbb{R}^n$ and a not open set in $\mathbb{R}^n$ be homeomorphic ? I guess the answer is no, but I can't prove it.
0
votes
0answers
30 views

$\underset{x\rightarrow x_0}{\lim}f(x)=y_0$ iff $\underset{n\rightarrow \infty}{\lim}x_n=x_0$ implies $\underset{n\rightarrow \infty}{\lim}f(x_n)=y_0$ [duplicate]

I have the following task: Let $(X,d)$ and $(Y,e)$ be metric spaces, $E\subset X$ and $x_0$ be an accumulation point of $E$. We say that point $y_0\in Y$ is the limit point of mapping ...
-1
votes
1answer
29 views

Homeomorphism between topological spaces defined by $f(x) < g(x)$

So, I have two continuous functions $f(x)$ and $g(x)$. $f,g : \mathbb{R} \longrightarrow \mathbb{R}$ and $f(x) < g(x)$ for all $x$ real. I have to show that $\{(x,y)\in \mathbb{R} | f(x) \leq y ...
1
vote
1answer
24 views

Connected spaces minus proper subspaces is connected

So, I have a topology problem here. It goes like this. We have X, Y conected topological spaces and A, B proper subspaces of X and Y respectively. I have to show that $X \times Y - A \times B$ is ...
0
votes
1answer
16 views

Number of connected components of boundary and interior

Let $A\subset \mathbb{R}^n$ be an open set, such that the boundary $\partial A$ has only finitely many connected components. Is it true, that $A$ can only have finitely many connected components as ...
0
votes
1answer
27 views

Equivalent Metrics on $\mathbb{R^n}$

I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces. Show that the following define equivalent metrics on ...
3
votes
0answers
52 views

Prove an annulus is homeomorphic to a cylinder

Let $A \subset \mathbb{R}^{2}$ be the annulus $A = \{(x,y) \in \mathbb{R}^{2} \colon 1 \leq x^{2} + y^{2} \leq 4 \}$. Prove that $A$ is homeomorphic to $S^{1} \times I$, where $I = [0,1]$ is the ...
0
votes
0answers
48 views

Looking for an example of a bounded set.

Consider the local base over the space of complex continuous functions over $[0,1]$ (denoted by $\mathcal{C}[0,1]$) defined for each fixed $x\in [0,1]$ and $\epsilon>0$ by ...
0
votes
1answer
37 views

Closed sets and accumulation points

In complex analysis how to prove that if $S$ is closed in $\mathbb{C}$ then it contain all of its accumulation points. If $S$ is closed then $S$ contain all its boundary points.(If $z_{0} $ is a ...
0
votes
2answers
53 views

If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected

If $\{E_\alpha\}_{\alpha\in A}$ is connected in $\mathbb{R}^n$, $\bigcap\limits_{\alpha\in A}E_\alpha \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E_\alpha$ is connected. I have zero intuition ...
5
votes
1answer
44 views

Definition of Sigma Algebra

I was wondering, why are we not allowed to take arbitrary unions (likewise intersections) in the definition of a sigma algebra?; I am looking for a more or less intuitive reason. It seems to me that ...
3
votes
2answers
40 views

Many point compactification

If $X$ is a noncompact LCH space (locally compact, Hausdorff) then its one point compactification is $X^*=X\cup \{\infty\}$ with topology $\mathcal{T^*}$ given by $U \in \mathcal{T^*}$ iff either a) ...
1
vote
2answers
36 views

If $A$ and $B$ are conneted and $A\cap B\neq \emptyset$, then $A\cup B$ is connected

Can you please let me know if my proof is reasonable? Prove: If $A$ and $B$ are conneted in $\mathbb{R}^n$ and $A\cap B\neq \emptyset$, then $A\cup B$ is connected Proof: Suppose that $A\cap B$ is ...
4
votes
2answers
61 views

Quotient space of the reals by the rationals

Let $\mathbb{R}/{\sim}$ be the quotient space given by the equivalence relation $a \sim b$ if $a$ and $b$ are rational. I am trying to understand general properties of the quotient topology and this ...
0
votes
1answer
27 views

The conjugation Group action is continuos

How can I prove that the group action from $G\times G\to G$ defined by $(g,x)\mapsto gxg^{-1}$ is a continuos function? I tried to use the known facts that multiplication and $(x,y)\mapsto xy^{-1}$ ...
0
votes
1answer
103 views

How should I prove the following? Algebraic topology and homeomorphism

I am struggling immensely with topology since the start of the course, probably due to its extremity; the explanations are either "very rough" or "very strict and rigid and hard to comprehend." Either ...
0
votes
1answer
45 views

Example of discrete set [closed]

Please I need examples of discrete set and non discrete set... am a little confused over this expression. I am thinking of discrete as a finite set but found from another article that, its not ...
1
vote
1answer
36 views

Show that if $H$ is a normal subgroup of $G$ then so is $\bar{H}$.

This is problem 4.14 in Armstrong's Basic topology: Let $G$ be a topological group. If $H$ is a subgroup of $G$, show that its closure $\bar{H}$ is also a subgroup, and that if $H$ is normal then ...
0
votes
1answer
90 views

What does the notation “*” mean?

I do not know the name of or what it does so I have no means of searching for an answer over the internet or a book. In my notes for algebraic topology, I have this bit that says, For any $f: X ...
3
votes
0answers
33 views

Open sets in $\mathbb{R}^{n+m}$ written as the union of $U \times V$ each open

Suppose I've got a set $Q \subseteq \mathbb{R}^{n+m}$, which is open in $\mathbb{R}^{n+m}$ (in terms of open balls). I wish to prove that there exist sets $\{U_{\alpha}\}_{\alpha \in A}$ and $\{ ...
1
vote
1answer
19 views

wrong proof of “locally lipschitz implies continuity”

I think that I've proved that locally lipschitz implies continuity on metric space. But something must be wrong: Let $(\mathfrak{X},d_1)$ and $(\mathfrak{Y},d_2)$ be metric spaces. If $\varphi ...
0
votes
1answer
50 views

Covering maps are proper?

Under wich conditions a covering map is also proper? For example the covering of the circle is clearly not proper Is there anything more general that say, when the cover is a compact space? Or having ...
1
vote
1answer
49 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
0
votes
0answers
29 views

Can one prove the existence of a fixed point for a shrinking map on a sequentially compact metric space WITHOUT proving the space is compact?

Let $(X,\rho)$ be a metric space with $Y\subset X$ a sequentially compact subspace, and a mapping $T:X\to Y$ satisfying $\rho(Tx, Ty)<\rho(x,y)$ for all $x\neq y$. Prove that $T$ has a unique ...
0
votes
0answers
25 views

Open sets in $n+m$ dimensions is the cartesian product of open sets in $n$ and $m$ dimensions

I'm considering the open sets of $\mathbb{R}^{n+m}$. I'm trying to show that any open set $Q \subseteq \mathbb{R}^{n+m}$, can be written as a cartesian product $Q = U \times V$, where $U$ is open in ...
1
vote
1answer
20 views

Show that every open set in second countable LCH space is $\sigma$-compact

Let $(X,\tau)$ be a second countable, locally compact Hausdorff space. Theorem: If $S \in \tau$, then $S$ is $\sigma$-compact. How do I show this statement? The following is what I have ...
4
votes
1answer
51 views

Homotopy equivalence between $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ and a discrete space

Consider the space $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ with the topology induced by the real line. Is $X$ homotopy equivalent to some enumerable discrete space $Y$? My try was the ...
1
vote
1answer
56 views

What's the precise mathematical theorem that's being cited when people write “if we let m go to infinity”?

I see this sentence used a lot but many times I'm confused by it. I imagine there's a theorem that's being implicitly cited every time this sentence is used. Could somebody please tell me which one it ...
0
votes
1answer
92 views

Interior point, limit point, isolated point, boundary point and cluster point

This is my first time take real analysis course and I think it is pretty hard. I think I have some difficulties to understand the definition for these points, I will try to explain what I am thinking, ...
0
votes
1answer
20 views

Showing an algebra separates points

Let $X$ and $Y$ be compact Hausdorff spaces. Show the algebra generated by functions of the form $f(x,y) = g(x)h(y)$, where $g \in C(X)$ and $h \in C(Y)$ is dense in $C(X \times Y)$ Call the ...
0
votes
1answer
22 views

Is part of a circle lying on first quadrant?

I have circle $ C: (x-x_0)^2+(y-y_0)^2\leq r^2$ with center $(x_0,y_0)$ and radius $r$. I want to find out in exactly what quadrants the circle lies. Is there a condition with this functionality? ...
0
votes
0answers
31 views

$G_\delta$ subgroups of a Polish Group

Let $X$ be a Polish Group. It's known that every its Polish subgroup is a $G_\delta$. Pick one of them, say $V$. Is it true that $V$ is the intersection of open subgroups? Thank you
0
votes
1answer
37 views

Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
1
vote
0answers
18 views

Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
1
vote
4answers
43 views

Introductory text on nets

I have learned point-set topology using filters. Now I do functional analysis where we are using nets to do topological stuff. Therefore I search an introductory text on nets that is suitable for this ...
0
votes
1answer
46 views

Proof check/ suggestion: The suspension of $S^n$

In one of my excercise sheets there was a remark saying that $$SX \approx S^{n+1}$$ where $SX$ denotes the suspension of $X=S^n$. So I tried to prove this on my own and would like to discuss my ...
1
vote
1answer
36 views

Degree of a non-surjective map f

In my notes I found an excercise claiming that $f: S^n \to S^n$ has $deg(f)=0$ whenever it's not surjective. I can prove this if I assume smoothness by applying Sard's theorem but I'm wondering if ...
1
vote
1answer
24 views

locally connected separable metric space but not completely metrizable

Can we find an example of a space which is locally connected separable metric space without isolated points but not completely metrizable?
1
vote
0answers
67 views

Can someone check my proof? (connectedness of real projective hyperquadrics)

Theorem: Let $Q_{\mathbb{R}} \subset \mathbb{P}^n_{\mathbb{R}}$ the set of real points of a projective hyperquadric. Prove that $Q_{\mathbb{R}}$ is connected with the topology induced by ...
1
vote
1answer
21 views

Distance function is continuous in topology induced by the metric

The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ ...
4
votes
1answer
52 views

Theorem 19.4 in Munkres' TOPOLOGY, 2nd ed: Does the converse also hold?

Let $J$ be an arbitrary (non-empty) index set, and let $\left\{ X_\alpha \right\}_{\alpha \in J}$ be an indexed family (or collection) of topological spaces, and let $\Pi_{\alpha \in J} X_\alpha$ ...
2
votes
1answer
62 views

Is this set open in the product topology?

Let $X$ and $Y$ be topological spaces and equipp $X\times Y$ with the product topology. Assume $U\subset X$ is open and for every $p\in U$ we have an open subset $V_p\subset Y$ of $Y$. Is the set ...
1
vote
1answer
18 views

Subset of first countable space closed iff intersection with every compact set is closed.

Let $(X,\tau)$ be a first countable Hausdorff space, and $S \subseteq X$. $S$ is closed iff every $S \cap T$ is closed, where $T$ is compact. The only if direction is trivial since every compact ...
1
vote
2answers
46 views

Interior and boundary of $\mathbb{Q}$ in $\mathbb{R}$

The closure of $\mathbb{Q}$ is $\mathbb{R}$, but what are the boundary and interior of $\mathbb{Q}$? I think both are $\mathbb{R}$ because in any open ball centered at $q\in \mathbb{Q}$, there is any ...
0
votes
1answer
24 views

on a countably union of $F_\sigma$ sets

Let $F$ be a countably union of $F_\sigma$ sets, say $F_i$ for $i\in \mathbb N$. Is it true that if $F$ is not meager then there is an $F_i$ with non-empty interior? For me it's true but I can't find ...
0
votes
1answer
6 views

on the interior of $F_\sigma$ sets

Let $X$ be a topological space. Let $F$ be an $F_\sigma$ set of $X$ such tat its closure has not empty interior. Is true that also $F$ has not empty interior? I tried with the known fact that the ...
1
vote
1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
0
votes
0answers
21 views

Non-convex open set whose closure is convex

In a topological vector space, can there be a non-convex open set whose closure is convex?
4
votes
2answers
48 views

Arzela-Ascoli for $\mathbb R^n$ from the case of $\mathbb R$?

In class, we proved the Arzela-Ascoli theorem for $\mathbb R$. The lecturer said it's also true for $\mathbb R^n$, and this version is deducible from $\mathbb R$. I tried to do this but failed. How ...