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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7
votes
1answer
48 views

Sphere homeomorphic to interval times space

Let $Y$ be any topological space. In my notes I found the exercise to show that: $I \times Y \approx S^n $ via a homeomorphism is not possible, where $S^n$ denotes the $n$-sphere and $I$ the unit ...
0
votes
1answer
38 views

Postnikov tower of a product

Let $X$ and $Y$ be simply connected, locally finite CW-complexes and let $(X_i)_i$ and $(Y_i)_i$ be their Postnikov towers respectively. Is the Postnikov tower of $X\times Y$ given by the products ...
0
votes
0answers
33 views

Relations between cluster points of nets and types of accumulation points of sets

Let $X$ be a topological space, $(x_\alpha)$ a net in $X$ and $A \subseteq X$ an arbitrary subset. The point $x \in X$ is a cluster point of $x_\alpha$ if for every neighborhood $U$ of $x$ the net ...
0
votes
1answer
37 views

Bijection bewteen $(-1,1)$ and $\{(x,y)\in\mathbb{R}^2:y=x^3\}$

I am trying to show that $(-1,1)$ and $\{(x,y)\in\mathbb{R}^2:y=x^3\}$ are homeomorphic, with the standard metrics. I cant see how to define a bijection.
1
vote
2answers
38 views

Definition of topological space

The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the ...
5
votes
3answers
94 views

Generalized Gauss map, giving rise to second fundamental form

I know that the tangent bundle of $G_n(\mathbb{R}^{n+k})$ is isomorphic to $\text{Hom}(\gamma^n(\mathbb{R}^{n+k}), \gamma^\perp)$, where $\gamma^\perp$ denotes the orthogonal complement of ...
1
vote
1answer
17 views

Sequentially compact iff every countably infinite subset has an infinite subset that has an $\omega$-accumulation point?

Let $X$ be a topological space and $A \subseteq X$ a subset. $A$ is called sequentially compact iff every sequence in $A$ has a convergent subsequence with limit in $A$. A point $x \in X$ is an ...
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 ...
2
votes
1answer
476 views

Show that any convex subset of $R^k$ is connected

I need to prove that any convex subset of $R^k$ is connected. I have seen the proof in Rudin's book and on numerous websites but they all use some prior results. I want to do it without using results ...
-1
votes
0answers
26 views

Why do we say a level 1 Menger Sponge has 5 holes? [closed]

I've heard that a level 1 Menger Sponge has 5 holes, but what is the justification for this? I can understand starting with a hole down the center, and making 4 more to meet it from the sides, but ...
1
vote
1answer
33 views

Are all the subsets of $\mathbb{Z}$ closed or open (or neither) in $\mathbb{Z}$?

At each integer $n$, $B_r(n)=\{n\}$ for small $r$, so $B_r(n)=\{n\} \subset \mathbb{Z}$. Since any subset is a union of some integers, does this imply that all subsets are open? Also, since there is ...
1
vote
1answer
53 views

Proving a topology is not induced by a metric

I'm reading a proof where it requires to show that a topology is not induced by a metric. My question is: What does it mean for a topology to be induced / not induced by a metric?
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
0answers
23 views

Is preimage of closure equal to closure of preimage under continuous topological maps? [duplicate]

Let $f:X \rightarrow Y$ be a continuous map of topological spaces and $B \subseteq Y$ Is it true that $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
5
votes
4answers
146 views

Topology $\text{i})$ What is a topology? $\text{ii})$ What does a topology induced by a metric mean?

I am now trying to understand what a topology and a topological space is. Yes, I know the "formal" or "mathematical definition" of it, it is in my notes so it's easy for me to reiterate that. Please ...
0
votes
1answer
32 views

Construction of a continuous function

Given two sets $x = \{ a_1, a_2, a_3, a_4 \}$ and $y = \{ \emptyset, x, \{ a_1, a_2, a_3\}, \{ a_3 \}, \{ a_3, a_4 \} \}$, where $y$ is a topology defined on $x$. How could we construct a continuous ...
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 ...
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
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
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 ...
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 ...
2
votes
1answer
55 views

Proving that if a function is a metric then it is symmetric and non negative

I am trying to prove that given a metric d using only the properties that it $d(a,b)=0 iff a=b$ and $d(a,c)\le d(a,b)+d(b,c)$ that $d(a,b)=d(b,a)$ and $d(a,b) \gt 0$ I understand that it is part 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.
1
vote
1answer
109 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 ...
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 ...
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
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 ...
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 ...
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 ...
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
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 ...
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 ...
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
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 ...
3
votes
2answers
255 views

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space?

Is there a first countable, 0-dimensonal, locally compact, lindelöf, non-compact space in which all non-empty open sets have $\pi$-weight $\mathfrak c$? It also can be seen here. Thanks for your ...
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 ...
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 ...
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 ...
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) ...
7
votes
3answers
494 views

Good metric on $C^k(0,1)$ and $C^\infty(0,1)$

What would be a good metric on $C^k(0,1)$, space of $k$ times continuously differentiable real valued functions on $(0,1)$ and $C^\infty(0,1)$, space of infinitely differentiable real valued functions ...
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 ...
5
votes
2answers
186 views

Connected subspaces

I guess there's something wrong with my thoughts about connectedness seen in a subspace of a topological space and I need your help. Let me explain: These are the definitions I have: A ...
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 ...
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
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 ...
11
votes
5answers
14k views

Limit points and interior points

I am reading Rudin's book on real analysis and am stuck on a few definitions. First, here is the definition of a limit/interior point (not word to word from Rudin) but these definitions are worded ...
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 ...