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
4answers
64 views

Prove $A=\{x\in \mathbb{R}|f(x)=x\}$ is closed subset of $\mathbb{R}$

Given $f\colon\mathbb{R}\rightarrow\mathbb{R}$ is continuous function. Prove that $A=\{x\in \mathbb{R}|f(x)=x\}$ is closed subset of $\mathbb{R}$. I totally do not have idea on how to start the ...
2
votes
0answers
18 views

Cut-number of Klein bottle and other non-orientable surfaces

What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these ...
2
votes
2answers
117 views

If two continuous maps into a Hausdorff space agree on a dense subset, they are identically equal

Let $f, g : X \to Y$ be continuous functions. Assume that $Y$ is Hausdorff and that there exists a dense subset $D$ of $X$ such that $f(x) = g(x)$ for all $x \in D$. Prove that $f(x) = g(x)$ for all ...
3
votes
1answer
39 views

Obtaining the Möbius strip as a quotient of $S^1\times[-1,1]$ by the antipodal relation

I am trying to obtain the Möbius strip as a quotient of $S^1\times[-1,1]$, where $S^1$ is, of course, the circle. My definition of Möbius strip is the quotient of the square $[0,1]\times[0,1]$ by the ...
2
votes
1answer
18 views

Alternative subbasis for the compact-open topology on $C(X,Y)$

Given spaces $X,Y$, where $X$ is Hausdorff, and the topology on $Y$ has basis $\mathcal{U}$, I would like to show that the set $\mathcal{S} := \{ S(K,U) \mid K \subset X, K \text{ compact, } U \in ...
0
votes
0answers
29 views

Compactness in space of continuous functions

Statement Let $C(X,Y)$ be the space of continuous functions endowed with a norm. Furthermore, we impose that $Y$ be a compact metric space. Question Does it necessarily follow that given a closed ...
2
votes
1answer
25 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
1
vote
1answer
18 views

The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
0
votes
0answers
9 views

Uniform continuity of scalar multiplication in topological vector spaces

If $X$ and $Y$ are topological vector spaces over $\mathbb R$, then a map $f:X\to Y$ is called uniformly continuous if for each neighborhood $V\subseteq Y$ of $0\in Y$, there exists a neighborhood ...
0
votes
0answers
10 views

ABout the closure of sequential spaces, and (non) existence of subsequence

I'm looking for a topological space $X$ a convergent sequence of functions $f_n$ such that $f_n\to f$ (puntually) and a family of succession $g_{m,n}$ such that $g_{m,n }\to_m f_n$ (puntually) but ...
3
votes
1answer
32 views

Some quotient of $[0,1]$ is homeomorphic to $[0,1] \times [0,1]$

I have been unable to solve this problem. I imagine you would start with some bijection $$f:[0,1] \rightarrow [0,1] \times [0,1]$$ But I don't get how any topological argument could be made here with ...
1
vote
0answers
19 views

$C([0,1], \mathbb R)$ with supperimum norm is not sigma compact [duplicate]

Assume the banach space $C([0,1], \mathbb R)$ with supperimum norm. I guess that this banach space is not sigma compact, in the other word $C([0,1], \mathbb R)$ is not union of countable compact ...
1
vote
1answer
27 views

What is the difference between a cover and a subset?

According to Wikipedia, A set $A$ is a subset of a set $B$ if A is "contained" inside $B$, that is, all elements of $A$ are also elements of $B$. A cover $C$ of a set $X$ is a collection of sets ...
2
votes
1answer
29 views

Showing homeomorphism between interval's quotient spaces

We have two spaces: $[0,1]/C$ and $[0,1]/A$, where $C$ denotes the Cantor set and $A=\{0,1,\frac{1}{2},\frac{1}{3},...\}$. One needs to show they are homeomorphic. What I thought about is showing ...
0
votes
0answers
36 views

Why quotient map maps open sets to open sets?

My book says that the quotient map of a topological space $X$ into its equivalence classes under some equivalence relation maps open sets to open sets.However my intuition tells me that this is wrong ...
0
votes
0answers
64 views

The hemisphere is contractible [on hold]

Please, where can i find the proof of the fact that The hemisphere is contractible ? I read that we must prove that the hemisphere is homotopoic to a disc, but how to find this homotopy? We ...
1
vote
2answers
36 views

$f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$

Let $f:S^{2n}\rightarrow S^{2n}$ continuous. Then there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$. I am having a hard time finding a starting Point. Thank you
0
votes
1answer
55 views

Which topology textbook has the greatest amount of ancillary support available on the Internet?

I'm considering to begin upon the study of topology and am wondering which book would the best option. I've even started reading Munkres and G. F. Simmons, but the problem is neither book has any ...
2
votes
1answer
32 views

Determining whether a given map is closed/a quotient map.

I would like to solve the following problem: Let $X$ be the union of the $x$- and $y$-axes in the plane and let $f$ be the function from $X$ to the $x$-axis that sends all of the $y$-axis to the ...
4
votes
3answers
94 views

Proving intersection of dense subsets of a metric space X is the isolated points of X.

Suppose X is a metric space. Let $\mathscr C$ denote the collection of all dense subsets of X. Show that $\bigcap\mathscr C $ = iso(X). Thus the question asks to prove that every dense subset of X ...
0
votes
1answer
36 views

Models of the hyperbolic plane

Hilbert's theorem tells us that there is no immersion in $\mathbb{R}^3$ with negative Gauß curvature that is complete. Despite, there are some models of surfaces with negative Gauß-curvature like the ...
3
votes
1answer
30 views

compact metric space where metric is surjective on $[0,1)$

I have the following problem and I was wondering if my argument below is okay: Let $(X,d)$ be a compact metric space with the property that for every $t \in (0,1)$, there exist points $x_t,y_t$ ...
0
votes
1answer
25 views

Questions about the space of rays with initial point the origin endowed with the quotient topology.

I need to know if some properties about the topological space $\mathbb{R}^n/{\sim}$ are true, where $\sim$ is a equivalence relation defined by $a\sim b \iff \exists \lambda >0$ such that ...
1
vote
1answer
82 views

Example of two closed continuous functions whose “product” is not closed

Let $X,Y,W,Z$ be topological spaces, and let $f: X \longrightarrow Y$, $g: W \longrightarrow Z$ be closed functions. Let $f \times g: X \times W \longrightarrow Y \times Z$ be such that $f \times ...
2
votes
0answers
28 views

ANR is locally contractible

Recall that a space $X$ is contractible if there exists a homotopy $h:X\times [0,1]\to X$ such that $h$ is equal to the identity map on $X\times\{0\}$ and $h$ is constant on $X\times\{1\}$. Please ...
0
votes
0answers
36 views

Number of topologies on a set

Let $X$ be a nonempty set with $n$ elements. I want to find an upper bound for the number of possible topologies for $X$. I proceed as follows: The power set $\mathcal P(X)$ contains $2^n$ elements. ...
2
votes
2answers
20 views

Property of distance and adherence

Please how to prove that in a metric space $(E,d),$ for $A,B\subseteq E$ that $\forall x\in E, d(x,A)=d(x,\overline{A})$ and that $d(A,B)=d(A,\overline{B})=d(\overline{A},\overline{B})$ and ...
1
vote
1answer
23 views

What can we say about two spaces quotiented by the same group action?

If $ X / U(1) \cong Y / U(1)$, what can I say about $X$ and $Y$ ? Will they also be isomorphic ? Here, $X$,$Y$ are two topological spaces and I chose $U(1)$ the circle action but it could be anything ...
0
votes
2answers
40 views

Covering space of $S^1 \vee S^1$?

Is this a covering space of $S^1 \vee S^1$? I'm not sure what the map from this space onto $S^1 \vee S^1$ does. What is mapped onto which $S^1$?
4
votes
1answer
101 views

Additive function $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ is zero everywhere.

Let $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ be an additive function ($f(x+y)=f(x)+f(y)$ for every $x,y \in \mathbb{Z}^\infty$). In addition for every $x=(0,\dots, 0,1,0, \dots)$ we have ...
5
votes
1answer
57 views

Do simply-connected open sets admit simply connected compact exhaustions?

Suppose $E$ is a simply-connected open subset of $\mathbb R^n$. Must there be a sequence of compact subsets $K_n$ such that $E = \bigcup_{n=1}^\infty K_n$, $K_n \subseteq K_{n+1}$ for all $n$, and ...
3
votes
1answer
64 views

Help understanding the argument used in this proof that $SO(n)$ is path-connected

Background: I am reading Tapp's matrix groups for undergraduates and I am in the process of showing that $SO(3)$ is path-connected. While working on my own arguemnt I found a proof online in these ...
2
votes
2answers
34 views

Categorical Pasting Lemma

If I'm not mistaken, the pasting lemma for a two-element open cover $X=A\cup B$ of a topological space is equivalent to saying that the following square is a pushout in $\mathsf{Top}$: ...
0
votes
1answer
19 views

Filters and Convergence

Show that ($\mathit{X}, \mathcal{T}_p$) -point chosen- is unique convergence except for the constant succession p (Where $\mathcal{T}_p$ is the point topology including). I've been reading an ...
2
votes
3answers
30 views

Question about adherence of sets

please I have that $A$ is compact and i want to prove that for a set $U$ we have $A= \overline{A\setminus U}\cap \overline {U}$ I do this: $x\in \overline{A\setminus U}\cap \overline ...
3
votes
0answers
33 views

Coequalizers are quotient maps

Is it possible to show that every coequalizer in the category of Hausdorff spaces is a quotient map directly from the universal property of a coequalizer and without use of the set-theoretical ...
0
votes
1answer
29 views

Definition of the support of a measure

Having read on Wikipedia about support of measure, I found two different definitions. The first one is in terms of measure spaces "All measurable sets should have nonzero measure". The second one is ...
5
votes
1answer
55 views

Totally disconnected space in which some quasicomponents have interior?

Assume all spaces are metric. Question. Does there exist a space $X$ which is totally disconnected (the components of $X$ are singletons), yet some quasicomponent of $X$ has nonempty interior? I ...
5
votes
2answers
96 views

For someone who is self-studying topology: what are the main topics to focus on?

I will have to teach myself topology for the Math GRE Subject Test because, although I graduated with a math major, I never took topology. I have Munkres and Kelley, along with the Schaum's Outlines ...
3
votes
1answer
47 views

Orientation preserving homeomorphism between half disk and waxing moon-shaped surface

I need to prove that the connected sum of two orientable (top.) manifolds is orientable. To do so, I need to find an atlas for the connected sum, and to find it, I need to provide an explicit ...
1
vote
1answer
41 views

Why is $\varphi\colon A^G\to A$ continuous?

Let $G$ be a group and $e_G$ its neutral element. Moreover, let $A$ be a finite set. $A$ is equipped with the discrete toplogy, $A^G$ with the product topology. Let $\tau\colon A^G\to A^G$ be ...
2
votes
2answers
117 views

Is a constant function between topological spaces continuous?

Let $T\colon X\to Y$ be constant, where $(X,\tau_1)$ and $(Y,\tau_2)$ are topological spaces. Maybe a silly question, but is then $T$ continuous? It is to show that for $O\in \tau_2$ I have that ...
4
votes
1answer
144 views

Open and closed sets of an infinite space

Let $X$ be an infinite set and $T$ a topology on $X$ such that each infinite subset of $X$ is closed. Show that $T$ is the discrete topology. The idea I have is to take two infinite subsets $A$, $B$. ...
1
vote
1answer
38 views

Fundamental group Pi1(SU(n)) and Pi2(SU(n))

I need to find the fundamental group $\pi_1(SU(n))$ and $\pi_2(SU(n))$ for all $n$. I don't have any idea.
1
vote
1answer
57 views

Verifying that a certain collection of intervals of $\mathbb R$ forms a topology

I'm doing exercise from "Topology without tears book", page 27, exercises a and b. First one (a) reads as follows. "Let $\mathbb{R}$ be the set of all real numbers. Prove that each of the following ...
2
votes
2answers
55 views

$[0,\omega_1]\times [0,\omega_1)$ is not $T_4$.

With $T_4$ I mean a normal and $T_1$ space. I know this result is true, in fact there exists the following general theorem: $X$ is paracompact iff $X\times \beta X$ is normal. Where $\beta X$ is the ...
5
votes
1answer
85 views

Subset of $\mathbb{R}^4$ such that the intersection with a hyperplane is dense and does not contains $4$ coplanar points.

Does it exist a subset $S$ of $\mathbb{R}^4$ such that for all affine hyperplane $H\subset \mathbb{R}^4$, the set $H \cap S$ is dense in $H$ and does not contains $4$ coplanar points? More than ...
2
votes
2answers
73 views

To determine if set is open or closed

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f (t) = t^{2} $ and let $U$ be any nonempty open subset of $\mathbb{R}$ Then a .$ f (U) $ is open b. $f (U) $ is closed c. $f^{-1}(U)$ is open ...
2
votes
3answers
269 views

A not complete metric space?

Please how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.
0
votes
1answer
22 views

Can continuity by sequences be applied?

I have a rather short question: If I have a compact topological space that is induced by a metric and I want to show continuity of a function on that topological space: Can I then use the criterion ...