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
2answers
16 views

To prove given set is basis for topology on $\mathbb{Z}$ [duplicate]

An arithmetic progression in $\mathbb{Z}$ is a set $A_a,_b=\bigg\{\dots,a-2b,a-b,a,a+b,\dots\bigg\}$ with $a,b\in\mathbb{Z}$ and $b\neq0.$ prove that the collection of arithmetic progressions ...
0
votes
1answer
29 views

Munkres Topology Article -30 Problem 5

Show that a metrizable space with a countable dense set has a countable basis. My try: Let $X$ be a metrizable space with a countable dense set $D$. Consider for each $n\in \Bbb ...
3
votes
4answers
42 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
0
votes
2answers
62 views

Is there a nice open set proof that multiplication is continuous?

For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the ...
0
votes
0answers
10 views

different definitions of a subnet

The classical definition of subnet seems to be that $\Psi: J\to X$ is a subnet of $\Phi: I\to X$ if there exists a monotone, final map $h: J\to I$ s.t. $\Psi = \Phi\circ h$. I found another definition ...
3
votes
2answers
32 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
1
vote
2answers
26 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
11
votes
4answers
674 views

How to define “being inside of something” in the context of topology?

I'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to ...
1
vote
1answer
30 views

$T^2-D$ does not retract to the boundary $\partial D$

First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of ...
0
votes
1answer
24 views

Product topology and projection mappings.

Let us consider two topological spaces $X$ and $Y$. Now let us consider projection mappings $p_1$ and $p_2$ defined from the product set of $X$ and $Y$ to $X$ and $Y$ respectively .The lecture notes I ...
0
votes
2answers
28 views

(i) $\{(x,y) \in \mathbb{R}^2 |\;xy = 1\}\,\bigcup\, \{(x,y) \in \mathbb{R}^2 |\;y = 0\}$ is not connected

I need to understand the following (i) $\{(x,y) \in R^2 |\;xy = 1\}\;U \{(x,y) \in R^2 |\;y = 0\}$ is not connected however (ii) $ Y = \{(x,y) \in R^2 |\;x^2 + y^2 < 1\}\;U \{(x,y) \in R^2 |\;y ...
1
vote
1answer
31 views

Homogeneous space minus a point

If $X$ is homogeneous and $p\in X$, then is $X\setminus \{p\}$ necessarily homogeneous? This seems to work with all the simple examples I've tried. I would be interested in any counterexamples. Or ...
0
votes
0answers
58 views

When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
2
votes
2answers
34 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
2
votes
1answer
20 views

Regarding part of proof of proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.

Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space. Part of Proof: Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity ...
16
votes
5answers
735 views

Is the set of all topological spaces bigger than the set of all metric space?

I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the ...
2
votes
2answers
26 views

Finding closures on $\mathbb R$ over a specific topology

I have the following topology over $\mathbb R$ $$ T = \{\emptyset\} \cup \{G\subseteq \mathbb R: \mathbb Q \setminus G \text{ is finite}\} $$ How could I study the closure of $\mathbb Q$ and $\mathbb ...
0
votes
1answer
29 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.
0
votes
0answers
19 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 ...
1
vote
2answers
37 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 ...
7
votes
1answer
46 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
37 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 ...
1
vote
1answer
14 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
31 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 ...
0
votes
0answers
25 views

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

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
50 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?
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 $f^{-1}(\overline{B})=\overline{f^{-1}(B)}$?
0
votes
1answer
31 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 ...
2
votes
1answer
16 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
14 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
24 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
17 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
0answers
47 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
35 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
100 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
34 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
51 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
29 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
51 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
46 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
0answers
30 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
50 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
42 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 ...