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
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1answer
42 views

Standard norm of $\mathbb{R}^3$

I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page: Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding ...
2
votes
1answer
93 views

Topologies of equivalent metrics

On $X= {\bf R}^2 - \{ (x_1,0)|\ x_1>0 \}$ define two metrics : $d(x,y) = |x-y|$ and $d_2$ is a path metric. Then $$ d(x,y) \leq d_2(x,y),\ d_2(x,y)\leq C(x,y)d(x,y) $$ Here if $x_n=(n,1/n),\ ...
0
votes
2answers
2k views

Prove that finite sets are closed

How would you prove that finite sets are closed? This is what I have now but it seems too simple: Let $A$ be a finite set where $A = \{x_1,x_2,\dots,x_n\}$ for some $n\in N$. Let $\hat A = \emptyset$. ...
7
votes
2answers
128 views

Continuity of a map with constrains

Let $A_i$ be a disjoint union of finite number of closed sub intervals of $[0,1]$, $1\leq i\leq n$. Each of $A_i$ has non-empty intersection with $A_j$. However, the intersection of each triple of ...
0
votes
1answer
246 views

Continuity of inverse on compact sets.

Problem: The inverse of a continuous injective function $f: A \rightarrow \mathbb{C}$ on a compact domain $A \subset \mathbb{C}$ is also continuous. My attempt: We want to prove that $f^{-1}$ is ...
2
votes
1answer
144 views

Understanding the inclusion of sets in the open category of X $Op_X$ and what \{pt\} denotes

What I am trying to understand is what is going on with the inclusion of sets, as if I understand correctly they are the morphisms of the category of open sets on X: $Op_X$ is the category of open ...
1
vote
3answers
71 views

For every $a\in [0,1]$ there is a subsequence converging to $a$.

Consider the sequence in $[0,1]$ given by $ \left( \frac{1}{2},\frac{1}{4},\frac{3}{4},\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8},\frac{1}{16},\frac{3}{16},\frac{5}{16},\ldots \right)$ Prove ...
4
votes
2answers
127 views

Show that for any open set $U$ in $\mathbb{R}^2$, $f(U)$ is an open set in $\mathbb{R}$?

Suppose $\mathbb{R}^2$ and $\mathbb{R}$ are topological spaces with standard topology. Let $f(x,y) = x + y^2$ . How do I show that for any open set $U$ in $\mathbb{R}^2$, $f(U)$ is an open set ...
27
votes
6answers
1k views

How to develop intuition in topology?

Is there any efficient trick (besides doing exercises) to develop intuition in topology? The question is general but i would like to add my view of things. I started to teach myself topology through ...
0
votes
2answers
42 views

Introductory Topology

$f, g:(\mathbb{R}, \tau_\mathbb{R}) \to (\mathbb{R}, \tau_\mathbb{R})$ are two continuous functions. Show that the set $\{x: f(x)\le g(x)\}$ is closed. I'm not really sure where to start this ...
1
vote
1answer
55 views

General Topology Question

$Y = [-1, 1]$ induced by the subspace topology from $\mathbb{R}$. $A = (-1, -1/2)\cup(1/2, 1)$ and $B = (-1, -1/2]\cup[1/2, 1)$. a) Are $A$ and $B$ open or closed in Y with the subspace topology? ...
1
vote
2answers
32 views

One point set in $[0,1]^{A}$ is not $G_\delta$ when A is not countable

I need to prove that one point set in $[0,1]^{A}$ is not $G_\delta$ when A is not countable I tried something like this: assume that $\{x\}$ is $G_\delta$ for some x $\in$ $[0,1]^{A}$ then $\{x\} ...
2
votes
3answers
221 views

Positive definite part of a symmetric matrix - or: are the positive definite matrices a retract of the set of symmetric matrices?

$\newcommand{\Sym}{\operatorname{Sym}}$ Denote by $\Sym(n)$ the set of symmetric, real $n\times n$ matrices and let $\iota:\Sym^+(n)\hookrightarrow \Sym(n)$ be the subset of positive definite ...
0
votes
1answer
85 views

Prove $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ are homeomorphic to each other

I have $\{(x,y) \in \mathbb R^2 | 0 < x^2 + y^2 < 1 \}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1 \}$ and need to prove they are homeomorphic to each other. I wanted to use the function ...
1
vote
3answers
87 views

Prove that $\{(x,y) \in \mathbb R^2 | y = x^2 \}$ is not compact

I know I need to choose an open cover and then show it has no finite subcover. If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?
2
votes
3answers
149 views

Is the compact interval $[0,1]$ in the usual topology compact in this new topology?

Let $\mathbb{R}$ be a topological space with topology consisting of the sets $A \cup B$, where $A$ is open in the usual topology, and $B \subseteq \mathbb{R} \setminus \mathbb{Q}$. Is the interval ...
2
votes
1answer
361 views

Equivalence of continuous and sequential continuous implies first-countable?

It is an immediate result that a map from a first-countable space is continuous iff it is sequentially continuous. I was wondering if the converse was also true. That is, is it true that if every map ...
0
votes
2answers
517 views

Separation of two disjoint closed subsets in a compact Hausdorff space.

Suppose $X$ is a compact Hausdorff space and $C,D$ are disjoint closed subsets of $X$. I want to show that there exist open disjoint $U,V$ with $C\subseteq U, D\subseteq V$. Since $C,D$ are disjoint, ...
2
votes
1answer
246 views

Prove that the fundamental group of $X$ is Abelian

Let $X$ be a path-connected topological space. And there is a continuous map $F: X\times X \to X$ such that: $$F(x,x)=x \ \text{ and }F(x,y)=F(y,x).$$ Prove: The fundamental group of $X$ is Abelian. ...
1
vote
2answers
30 views

Need help in this hypothesis in general topology

Is it true that if every finite subset of a topological space $X$ is closed and every subset of $X$ is compact, then, $X$ has the discrete topology. I couldn't disprove or prove this statement and I ...
0
votes
1answer
39 views

Infinite subset of a compact topological space

I just can't quite get this question: Let $X$ be a compact space, $B_{n}$, n $\in \mathbb{N}$ a closed non-empty subset such that $B_{n+1} \subseteq B_{n}$. Show that $$\bigcap_{n=1}^{\infty} B_{n} ...
4
votes
1answer
50 views

Noetherian topological subspaces

I'm trying to prove that any subset of a noetherian topological space is noetherian in its induced topology. MY ATTEMPT OF SOLUTION Let $X$ be a topological space and $Y$ a subspace of $X$. If ...
5
votes
0answers
121 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
1
vote
1answer
176 views

Connected components of a topological space and Zorn's lemma

I have just come across the idea of a connected component of a topological space. And firstly I would just like some clarity on the definition, as it seemed a little vague. Here is what I understand ...
0
votes
1answer
31 views

Existence of a norm

K - compact, convex subset of $ \Bbb R^n $ 0 $\in$ int K K is symmetrical to 0. I'm sorry, but i don't know how to write it properly. I mean: $ (x_1,x_2,...,x_n) \in K \Rightarrow (-x_1,-x_2,...,-x_n) ...
1
vote
1answer
125 views

Hausdorff Non-First-Countable Quotient Spaces of Ordered Square

My question is whether or not there exist any quotient spaces of the ordered square $[0,1]^2$ such that it is both Hausdorff and is NOT first-countable. Attempts at the Solution: My first ...
0
votes
3answers
231 views

Help with Pointwise and Uniform Convergence in Metric Spaces

I am having a bit of difficulty understanding uniform convergence and would also like to check my understanding of pointwise convergence. Using the example of $f_n$(x) = $x^n$ on (-1,1), I found the ...
2
votes
1answer
77 views

Infinite dimensional euclidian space with the product topology metrizable?

Let $\mathbb{R}^{\omega}$ be the space of real sequenes with the product topology. Is $\mathbb{R}^{\omega}$ metrizable?
1
vote
0answers
17 views

Count the number of topological sorts for each poset [duplicate]

Count the number of topological sorts for each partially ordered set $(A,|)$, where (a) $A = (3, 5, 7, 11, 13, 16, 17)$ (b) $A = (1, 3, 9, 27, 81, 243)$ That is, you have to find the number of ways ...
4
votes
2answers
50 views

Open subsets of the closure

I want to prove that every open subset of a topological subpace is an open subset of its closure. Let $Y$ be a topological space and $X$ a subspace of $Y$. If $U$ is an open subset of $X$, we have ...
1
vote
1answer
98 views

Urysohn's Lemma and Normal space

Assume that $X$ is $T_1$-topological space such that, for all pairs of closed sets $A,B\subset X$ with $A\cap B$, there exist a continuous function $f:X\rightarrow [0,1]$ with $f(x)=0$ for all $x\in ...
1
vote
1answer
102 views

Quotients of the Ordered Square

Let $[0,1]^2$ be the ordered square; i.e. it has the order topology given by the dictionary order. This is a first countable compact space. Let $\Delta=\{(x,x)\mid x\in [0,1]\}$. Then is the ...
0
votes
1answer
46 views

$T_1$space on cocountable topology

Let $X$ is uncountable set with cocountable topology. is $(X,T)$ a $T_1$space? I tried to solve it in following way but I don't know it is correct or no? Is it sufficient? I know this topology is ...
5
votes
1answer
389 views

Compact metric spaces is second countable and axiom of countable choice

Why we need axiom of countable choice to prove following theorem: every compact metric spaces is second countable? In which step it's "hidden"? Thank you for any help.
7
votes
7answers
464 views

Enjoyable book to learn Topology.

I believe Visual Group Theory - Nathan Carter is the best book for a non-mathematician (with high school math) to learn Group Theory. Could someone please recommend me a similar book (if there is) to ...
1
vote
1answer
42 views

Prove that x is in the boundary of A iff x is an accumulation point of the complement of A given x is an isolated point of A

The complete question is the following: "Let A be a subset of metric space X and let x be an isolated point of A. Show that x is in the boundary point of A iff x is an accumulation point of $A^C$. I ...
0
votes
1answer
53 views

I didn't understand this open disk question

I don't understand why I can't connect the $-1$ and $1$ points with just two line segments. I've tried it in my head and it makes sense to me. Why do I need $3$ line segments? Can somebody draw this ...
2
votes
2answers
226 views

Prove that the x-axis in R^2 with the Euclidean metric is closed

I want to show that the x-axis is closed. Below is my attempt - I would appreciate any tips on to improve my proof or corrections: Let (X,d) be a metric space with the usual metric. WTS: {(x,y) | X ∈ ...
2
votes
3answers
156 views

Prove $\{(x,y) \in \mathbb R^2|x^2 + y^2 > 1 \}$ is not simply connected

I have literally no idea how to do this. My assignment question asks me to prove that $\{(x,y) \in \mathbb R^2|x^2 + y^2 > 1 \}$ and $\{(x,y) \in \mathbb R^2|x^2 + y^2 < 1 \}$ are homeomorphic ...
3
votes
1answer
109 views

Is there anything wrong with this use of the axiom of choice?

I have a proof of the following theorem: Let A be a finite set and X be a perfectly normal topological space, and let $\{(A,\succsim_x): x\in X\}$ be a family of binary relations on $A$ satisfying ...
4
votes
1answer
69 views

Is a graph $G$ completely determined (up to labelling) by its spanning trees?

The title is essentially the question. I know that trees can be represented as a topology (equivalently a topological closure operator) on a set -- so I'm wondering if the collection of spanning trees ...
0
votes
2answers
338 views

Are $\{(x,y) \in \mathbb R^2 | x^2 + y^2 < 1\}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1\}$ homeomorphic?

I have two sets $\{(x,y) \in \mathbb R^2 | x^2 + y^2 < 1\}$ and $\{(x,y) \in \mathbb R^2 | x^2 + y^2 > 1\}$ and need to prove they are homeomorphic. I believe I can use the function $f(x,y): ...
1
vote
1answer
134 views

What is an “essential loop”?

I'm a bit confused. Is an essential loop in a topological space $X$ just a loop $\alpha$, which is not-contractible (i.e. $[\alpha] \neq 0$ in the fundamental group of $X$), or is there something more ...
0
votes
1answer
44 views

Cardinality of subbasis for topological space

Claim: If $(X,\tau)$ is a topological space, $\mathcal B$ a base for $\tau$ and $\mathcal U$ an open cover of $X$ then there is a subcover $\mathcal V \subset \mathcal U$ whose cardinality is not ...
0
votes
1answer
81 views

How to show the set F of all finite sequences is connected in the space c0?

Question: How one can show that the set F of all finite sequences (i.e after n, the entries are zero) is connected in the space c0 (i.e. the space of all sequences that converge to zero) the metric ...
1
vote
1answer
45 views

isomorphisms- subspaces in topology

Consider the following topological spaces: $(X_1,\tau_1)=(\Bbb R,\tau_u)$ and $(X_1,\tau_2)=(\Bbb R, \tau_{kol})$ So the product topology is the following: $(\Bbb R^2, \tau_u \times \tau_{kol})$ I ...
2
votes
1answer
66 views

Topology Homeomorphism

Let $R$ be endowed with the subspace topology inherited from the euclidean topology. Let $A$ and $B$ be two subsets of $R$, where $A=\{3\}∪[4,5]∪\{6\}$ and $B=[3,4]∪\{5\}∪\{6\}.$ How do you show that ...
2
votes
0answers
47 views

What is the name for the topology where every point is in the boundary of an open set?

Is there a name for topological spaces in which every point is in the boundary of an open set?
6
votes
1answer
118 views

Uncountably many non-homeomorphic compact subsets of the circle

As the title says, the question is whether there are uncountably many non-homeomorphic compact subsets of the unit circle. I'm assuming this is true, but I wouldn't mind an elegant proof.
1
vote
1answer
43 views

Does $T_1$ imply Fréchet–Urysohn (every limit point is a limit of some sequence)?

It's not a problem from a book so I’m not even sure the statement is true. Nevertheless here's an alleged proof: ADDED LATER: Although the result is wrong I can't find a problem with the proof. I ...