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
0answers
14 views

Existence of (non) complete metric on an interval

I am stuck with this problem. Can anyone help me out? Thank you in advance. Which one of these are correct. a) (0,1) with the usual topology admits a metric which is complete. b) (0,1) with the ...
3
votes
2answers
23 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
1
vote
1answer
34 views

What does neighborhood/ball/closure mean in a non-metric and/or finite space?

I'm trying to understand these fundamental concepts of topology. I understand what open closed and boundary mean in a continuous space with a metric, but I'm having issues understanding the meanings ...
0
votes
1answer
23 views

Is the Odd-even topology weakly countably compact?

Something eludes me in the proof that the odd-even topology is weakly countable compact found here : https://proofwiki.org/wiki/Odd-Even_Topology_is_Weakly_Countably_Compact I don't understand why ...
1
vote
2answers
27 views

Question about proof of the tube lemma for metric spaces

Tube lemma: Let $M$ be a metric space and $K$ a compact metric space. Let $a\in M$, $a\times K\subset V\subset M\times K$, that is, suppose there is an open set $V$ between $a\times K$ and $M\times K$...
1
vote
0answers
12 views

What is the difference between Yoneda and Smyth-completeness?

What is the difference between Yoneda and Smyth-completeness. I know that Smyth-completeness is a stronger property than Yoneda-completeness; however, I was unable to find a simple definition for both ...
0
votes
2answers
32 views

A theorem about one-dimensional convex sets

Suppose we have a non-empty convex set which does not consist of only one point such that it belongs to the same line, then this set is either a line segment(closed, half-open or open),a ray(closed or ...
0
votes
1answer
21 views

Perfectly Normal is hereditary

The definitions I'm working with: $(X, T )$ is called perfectly normal if whenever $C$ and $D$ are disjoint, nonempty, closed subsets of $X$, there exists a continuous function $f : X \rightarrow [...
3
votes
2answers
107 views

Question on inverse limits

1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every $n\in\mathbb{N}$. Let $\...
0
votes
0answers
13 views

Dirac functional embedding

I got the following set up: Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...
2
votes
1answer
31 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
2
votes
0answers
22 views

Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$.

Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$. Let $n \in \mathbb N$ then we know $(n - \frac{1}{2}, n + \frac{1}{2})$ is open in ...
3
votes
3answers
41 views

If $(X,\tau)$ has no open connected set, must $(X,\tau)$ be totally disconnected?

Let $(X,\tau)$ a topological space and suppose that for all open sets $U \in \tau$ we have that $U$ is disconnected. Can we conclude that $(X,\tau)$ is totally disconnected, i.e. its connected ...
0
votes
1answer
20 views

Find the Limit point of this exercises

Good morning, i'm working in this exercise and i solve this, but, i don't know it's fine, please how you can find the limit point? 1) $\left\{ 1-\frac{1}{n}\::\:n=1,2,3...\right\}$ Well, i say the ...
2
votes
2answers
43 views

Map from 2-sphere into $(\mathbb R^3, |\cdot|)$

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. Since $S^2$ is compact, $T$ ...
0
votes
2answers
48 views

What is embedding?

I am new to this so do I need to learn topology in order to understand this? Cause I come across this which says that unlike the 2D sphere, 2d saddle surface cannot be embedded in 3D Euclidean space(...
0
votes
3answers
41 views

Prove or disprove $A$ compact/closed $\implies$ $\mathcal{P}(A)$ compact/closed

For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by $$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$ Prove or disprove that $A$ ...
9
votes
2answers
113 views

Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems ...
-1
votes
2answers
40 views

It's given for $\left \{ A=\frac{1}{n},n\in \mathbb{N} \right \} $=${1,\frac{1}{2}…}$. Why is $0$ accumulation point?

It's given for $\left \{ A=\frac{1}{n},n\in \mathbb{N} \right \} $$={1,\frac{1}{2}....}$. Why is $0$ accumulation point? It's never going to reach 0 because it's not in $\mathbb{N}?$
-2
votes
0answers
44 views

$X := \prod_{i\in I} X_i \: \: $ Show that X is (path-)connected, if $X_i$ is (path-)connected $\forall i \in I$

Let $I$ be an indexset and $(X_i, \mathcal T_i)$ a topological Space for $i\in I$. Let $X = \prod_{i\in I} X_i$ have the product Topology. Now i have to show the following two things: $X$ is ...
4
votes
3answers
155 views

Extension of identity map

Suppose $z$ be the identity map on unit circle $\Bbb T$. Extending $z$ to closed unit disc $\Bbb D$. The identity map on $\Bbb D$ is an extension of $z$. Is the continuous Extension of this certain ...
1
vote
2answers
34 views

embedding a T1 space into a product topology

is the following statement true or false: Let $X$ be a $T_1$ topological space, there exists a cofinite topological space $Y$, such that $X$ can be embedded into ${Y}^J$
1
vote
0answers
20 views

set equalities proofs

I'm teaching my self topology with the aid of a book, the problem i'm on ask to prove the following: Let $X$ be a topological space and $B$ be a subset of $X$. Prove the following set equalities.(...
1
vote
0answers
26 views

Verify if this is correct idea of continuous and homeomorphism

This is a follow up to a previous question. My main motivation is to understand these defintions: Definition of homeomorphism: If $X$ and $Y$ are topological spaces, a homeomorphism from $X$ to $...
0
votes
2answers
51 views

$\phi:M\to \mathbb{R}$ continuous, $\phi(x)<\epsilon$ for $x\in X$, then $\phi(x)\le \epsilon$ for $x\in\overline{X}$

I was reading a proof that if a sequence of functions from $M$ to $N$, where $N$ is complete, converges uniformly in $X$, then they converge uniformly in $\overline{X}$, and it uses this result: $\...
1
vote
0answers
12 views

Two questions on completely regular filters in locales

I'm reading the exposition of the Stone-Čech compactification for locales in Johnstone's book Stone Spaces. In Chapter IV Paragraph 2.2, Johnstone constructs the Stone-Čech compactification of a ...
0
votes
1answer
33 views

Compact metrizable space is separable proof question.

Prove that a compact metrizable space is separable. I am confused by a specific case of a compact metrizable space. Let $[0,1]$ be a compact metrizable space. Since $[0,1]$ is metrizable, it's ...
2
votes
1answer
62 views

if $M$ is compact, then every continuous bijection $F:M\to N$ is an homeomorphism

My book proves that: if $M$ is compact, then every continuous bijection $f:M\to N$ is an homeomorphism by the following: Being $f$ closed, your inverse $g:N\to M$ is a function such that $F\subset ...
1
vote
1answer
32 views

elements of topology is open set?

Let's consider the topological space $(\mathbb{R},2^\mathbb{R})$, then each interval is element of topology $2^\mathbb{R}$, but as usually, we consider $[a,b]$ as as closed set. So my question is ...
2
votes
0answers
34 views

Prove $\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1 dx+ \mathcal O(\int_\Omega1 dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2)$?

Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$. Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\...
0
votes
0answers
13 views

A continuum X is hereditarily indecomposable iff C(X) is uniquely arcwise connected

I have to prove as for my homework that a continuum $X$ is hereditarily indecomposable iff $C(X)$ is uniquely arcwise connected, where $C(X)$ denotes the hyperspace of all subcontinua of $X$. Here's ...
2
votes
2answers
30 views

two metrics on X such that lim d1(xn,x)=0 <=> lim d2(xn,x)=0, does it imply the identity of the two induced topologies?

Two metrics $d_1, d_2$ on $X$ For all $x_n, x$ from $X$ it holds: $$\lim d_1(x_n,x)=0 \iff \lim d_2(x_n,x)=0$$ Does it imply that the topology induced by $d_1$ is the same as the topology induced by $...
1
vote
0answers
29 views

Question about proof ot Tychonoff's theorem for metric spaces

Tychonoff's theorem: The cartesian product $M = \prod_{i=1}^{\infty}M_i$ is compact $\iff$ each $M_i$ is compact. My book, before proving it, says that the proof will happen like this: Given an ...
3
votes
2answers
35 views

2-Sphere is connected

I am supposed to show, that if I have a continuous function $F:S^2→ (\mathbb{R}, | · |)$, with $S^2$ being the 2-Sphere in $\mathbb{R^3}$, that there is a point where $F(x)=F(-x)$. Now I know how to ...
2
votes
2answers
30 views

Question about Definition of homeomorphism (counter example)

I'm teaching my self topology with the aid of a book, but i'm confused about the meaning of homeomorphic. Below, I have 2 topologies, $\mathscr{T}_1$ and $\mathscr{T}_2$ and I'm pretty sure they are ...
0
votes
2answers
64 views

$M\times N$ compact $\implies$ $M$ compact and $N$ compact

I must prove that $M\times N$ compact $\implies$ $M$ compact and $N$ compact using the definition that, if a metric space $M$ is compact, then every cover has an open finite sub cover. $$M=\cup ...
1
vote
1answer
20 views

Showing a space is homeomorphic to the Hilbert cube.

Let $(X_i,T_i)$ be a countably infinite family of topological spaces each of which is homeomorphic to the Hilbert cube. Show that $\prod_{i=1}^{\infty}(X_i,T_i) \cong I^{\infty}$. The question also ...
2
votes
2answers
46 views

Theorem 2.17 from RCA Rudin

I understood the proof of points $(a)$ and $(c)$. But I can't understand the proof of $(b)$. It's obvious that every closed set is $\sigma$-compact. But how Rudin applies $(a)$ here? We have to show ...
2
votes
1answer
39 views

Is $(X,\mathcal T)$ a $T_0$-space?

Let $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ be topological spaces. Now define $\mathcal T=\mathcal T_1 \cap \mathcal T_2$. If $(X,\mathcal T_1)$ and $(X,\mathcal T_2)$ are $T_1$-spaces, is $(...
0
votes
2answers
29 views

Proving this quotient space is a Hausdorff space

Define $S^1 = \{x \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$. Define the equivalence relation $\sim$ as follows: $(x,y) \sim (x',y')$ if and only if $y = y'$. Now prove that the quotientspace $X/\sim$ with ...
2
votes
1answer
24 views

X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to exactly one on your list.

I'm teaching my self topology with the aid of a book. I'm trying to do the following problem: Let X={1,2,3}. Give a list of topologies on X such that every topology on X is homeomorphic to ...
2
votes
1answer
21 views

Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
1
vote
1answer
23 views

Show excluded point topology is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_4=\{U \...
3
votes
1answer
34 views

Connectedness of Product Topology

I am working on a pretty straight forward proof, I am trying to show that when I have a family of topological spaces $ (X_i, T_i )_{i \in I}$ where all $(X_i, T_i )$ are connected that the product ...
1
vote
2answers
23 views

Show particular point topology, is a topology

I'm teaching my self topology with the aid of a book. I'm trying to prove the following is a topology: Let X be an infinite set, and $p$ be an arbitrary point in $X$. Show that $\mathscr{T}_3=\{U \...
0
votes
1answer
26 views

Confusion about difference between Normal and Perfectly Normal

I am working with the following definitions: A topological space is normal if and only if every pair of disjoint, nonempty closed sets can be separated by a continuous function. A ...
0
votes
0answers
20 views

Prove that the cantor space is totally disconnected.

Prove that the cantor space is totally disconnected. Let $(G,T)$ be the Cantor space and let $\prod_{i=1}^{\infty}(A_i,T_i)$ be homeomorphic to the Cantor space where $(A_i,T_i) = (\{0,2\}, T_{...
1
vote
4answers
41 views

Correctness of proof that every neighborhood is an open set.

Rudin makes the following definitions: (a) A neighborhood of p is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$, for some $r > 0$. (b) $E$ is open if every point of $E$ is an ...
0
votes
1answer
36 views

Baire space using extended metric?

Consider the set $C^1(\mathbb{R},\mathbb{R})$ of continuously differentiable functions on $\mathbb{R}$, endowed with the extended $C^1$ norm $\|f\|_{C^1} = \sup_{x\in \mathbb{R}} |f(x)| + \sup_{x\in \...
2
votes
2answers
56 views

Consequence of Riesz Representation Theorem from Rudin RCA

It's Riesz Representation Theorem from Rudin's book. In the following chapter I met the following example: It's obvious that $\sigma$-compact set has the $\sigma$-finite measure. But how to prove ...