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; ...

learn more… | top users | synonyms (3)

2
votes
1answer
17 views

Continuity on the parameters of the intermediate value theorem

Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that $F(x, 0) ...
1
vote
1answer
37 views

Prob. 10 (d), Sec. 19 in Munkres' TOPOLOGY, 2nd ed: How to show that this map is open?

Here's Prob. 10, Sec. 19 in the book Topology by James R. Munkres, 2nd edition: Let $A$ be a set; let $\{X_\alpha \}_{\alpha \in J}$ be an indexed family of spaces; and let $\{ f_\alpha ...
0
votes
1answer
16 views

Tychonoff space with unique compactification and 3 disjoint non-compact closed subsets

Prolog : The only compactification of a non-compact normal space $S$ is the one-point (Alexandroff) compactification IFF whenever $A,B$ are disjoint closed subsets of $S$, at least one of $A,B$ is ...
0
votes
0answers
25 views

What does the “closure of its graph” mean

I Am confused with various terminologies spelled out the same but meaning very differently depending on the situations. There are just too many. Here, I only understand the "closure" in the ...
1
vote
1answer
27 views

Is $\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$ locally connected?

Let $X=\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$. Determine whether or not $X$ is locally connected and find its components. Well, I know that a space $X$ is said ...
0
votes
0answers
18 views

Show that $h$ is homotopic to the identity map relative to $C$.

This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3. 3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar ...
0
votes
1answer
72 views

Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
2
votes
0answers
36 views

How to prove this criteria of differentiability? [duplicate]

Let $f: I \to \mathbb{R}$ continuous and $a\in \operatorname{int}(I)$. Suppose that there is $L\in\mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{ y_n-x_n}=L$$ for all sequences $(x_n)$ and $(y_n)$ ...
1
vote
1answer
105 views

Subsets of the reals when the Continuum Hypothesis is assumed false

If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary ...
1
vote
0answers
35 views

Proving open neighbourhood in topology

Let $X$ be the set $(\mathbb{R}\backslash \mathbb{N}) \cup \{1\}$. Define a function $f:\mathbb{R} \rightarrow X$ by $$ f(x) = \left\{ \begin{array}{ll} x & \mbox{if $x \in ...
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
43 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
44 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
19 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
50 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
89 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
30 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
17 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 ...