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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4
votes
2answers
81 views

Totally disconnected topologies on countable set.

Are there totally disconnected topologies $\tau$ on a countable set $X$ such that $(X,\tau)$ is not homeomorphic to one of the following? $\mathbb{N}$ with the discrete topology; one-point ...
2
votes
2answers
29 views

a question about connected set, how to know whether A is connected or not?

In the Euclidean plane $R^2$,consider the subset $$ A=\{(x,y)\in \Bbb R^2|\text{Either $x$ or $y$, but not both, is a rational number}\} $$ Is $A$ connected? Is $\Bbb R^2$\A connected? I have ...
0
votes
1answer
39 views

Is the definition of continuity in analysis a particular case of topological continuity?

Take a constant function and remove an open interval from it: $$f(x)= 1, \text{if $x\in(-\infty,0]\cup[1,\infty)$ }$$ This function shouldn't be continuous because at $0$ no right limit of the ...
4
votes
2answers
48 views

Show that the set of all complex numbers $z$ such that $|z| \leq 1$ is closed?

I'm working through Rudin's "Principles of Mathematical Analysis" on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rudin states without proof that the set $X = ...
3
votes
0answers
32 views

Topological proof for this set theory statement

Let $\mathcal{A}$ be an algebra of set (in a space $X$), such that any subcollection of disjoint sets in $\mathcal{A}$ is finite. Prove that $\mathcal{A}$ is finite. I already found a boring brute ...
0
votes
0answers
16 views

Exhaustion of a manifold by compacts

I searched for a proof of the following statement, but did not find one. I want to check if a proof I made is correct, or if I'm leaving out some detail and/or complicating things: Proposition: ...
2
votes
0answers
38 views

A question about compact sets: how to prove $g$ must be an isometry

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
0
votes
2answers
50 views

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic?

Are $[0,1)\times [0,1)$ and $[0,1]\times [0,1)$ homeomorphic? Not getting any idea how to start.
0
votes
1answer
39 views

What is the largest complete subspace of $(\mathbb{Q}, |\cdot|)$

For example $\left\{\frac{1}{n}\right\}\cup \{0\}$ is a complete subspace of $\mathbb{Q}$, but I am having trouble writing out the largest (in the sense of "$\subset$") complete subspace in ...
1
vote
1answer
19 views

a question about compact set, how to prove there exits f(y)=y [duplicate]

Let (X,p) be a compact metric space.Suppose that f X->X is a function such that, for all $x_1$,$x_2$ $\in$X, if $x_1\neq x_2$ then p(f($x_1$),f($x_2$))<$p($x1$,$x2$)$. Prove that there exits a ...
3
votes
2answers
40 views

One point compactification of $\Bbb{R}\setminus \{0\}$

What will be one point compactification of $\Bbb{R}\setminus \{0\}$? It looks like it will be union of two circles touching at a point. But do I write a Mathematical proof to justify my claim?
0
votes
2answers
20 views

$x$ x $1/x$ for $\epsilon$ $\gt 0$ has no $\epsilon$-neighborhood in $R_{+}$ x $R_{+}$

This is a problem from Munkres' Topology. Define the $\epsilon$-neighborhood of $A$ in a metric space $X$ to be the set $U(A, \epsilon) = ${$x$ | $d(x,A)$ $\lt$ $\epsilon$}. (d) Assume that $A$ is ...
1
vote
1answer
27 views

sequence of close and bounded sets in a prefect space

Suppose that$(E_n)$$_{n \in \mathbb N}$ be a sequence of closed and bounded sets in complete space $M$ such that $ E_{n+1} \subseteq E_n$ for all $ n \in\mathbb N$. If $\lim \operatorname{diam} E_n ...
1
vote
2answers
39 views

Frechet-Hausdorff theorem reference from J.L. Kelley used in proof that each probability measure is inner regular

Theorem: If $S$ is a complete, separable metric space, then each probability measure on it is inner regular. Proof: Since $S$ is separable, for each $n \in \mathbb{N}$ there exist countably many ...
0
votes
3answers
38 views

Let (X, d) be a metric space and A, B ⊂ X be two compact subsets. Show that A ∩ B is also compact

Question seems fine i just have a few doubts. Is it possible to just use the Heine Borel theorem? as both A and B are compact it implies they are both closed, so therefore their intersection is ...
1
vote
0answers
17 views

Is the translation of open and closed sets to some language non-antonym preserving?

Maybe more than one person though, before you were given the definition of closed set, that they were the sets that are not open, i.e. that the property of open and closed being antonyms were ...
0
votes
1answer
20 views

what is the definition of “two parallel copies of a surface S”

As indicated in the title, suppose $S$ is a surface with genus $g$, then what is the definition of "two parallel copies of S"?
1
vote
1answer
23 views

A problem in compactness in Euclidean Space using a special topology

Let $\mathscr U$ denote the usual topology on $R^2$ and consider the topology $\mathscr T = ${$U$ $\subset$ $R^2$ | $R^2 - U$ is a compact subset of ($R^2$, $\mathscr U$)} $\bigcup$ {$\emptyset$}} ...
1
vote
0answers
59 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
1
vote
0answers
12 views

Seperation of convex compact sets with affine halfspaces

Let $C_1,C_2,...,C_m$ compact convex sets s.t. $\bigcap C_i = \emptyset$. I want to show that in that case there exsist affine halfspaces $H_i$, such that for every $i=1,2...,m$, $C_i \subset H_i$ ...
1
vote
2answers
27 views

$\phi, \psi$ homeomorphisms on $U, V$ $\implies$ $\phi(U\cap V) \cong \psi(U\cap V)$?

Let $U,V \subset M$ be open subsets in some manifold $M$. Let $\phi, \psi$ be homeomorphisms on $U, V$ respectively. Is it true that we then have $\phi(U\cap V) \cong \psi(U\cap V)$?
-1
votes
2answers
20 views

Continuity of addition map with lower limit topology

Prove $f:\Bbb R\times\Bbb R \to\Bbb R$ (with lower limit topology on $\Bbb R$ in range and product topology on $\Bbb R\times\Bbb R$ from $\Bbb R$ with lower limit topology), where $f((x,y)) = x+y$, is ...
1
vote
2answers
46 views

Prove there exists dense open set

Let $G$ be an open set in $X$ and $D$ be a dense open set in $G$.Show there exists a dense open subset $V$ of $X$ such that $V\cap G=D$. Since $D$ is open in $G$, there exists $V$ open in $X$ ...
0
votes
1answer
48 views

Topological , Homeomorphic version of $|S \times S|=|S| $

Give example of a subset $A$ of $\mathbb R$ such that with respect to some topology , $ A$ is homeomorphic to $A\times A$ . In set theory ZF it is known to be equivalent to A.C. that for any ...
5
votes
1answer
117 views

algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
0
votes
2answers
36 views

If two sets are separated, then any two subsets of those sets are also separated?

I want to prove that if two sets X and Y are separated, then subsets of those sets are also separated. The definition is that if X intersect Y closure is empty and X closure intersect Y is empty, the ...
1
vote
2answers
31 views

Closed set on topological space [duplicate]

This is a problem on topological spaces and continuous functions. If $f,g \to\mathbb{R}$ are continuous functions, then $T=\{x\in X: f(x)=g(x)\}$ is closed on X
0
votes
0answers
25 views

Question about dimension in Notherian spaces

Let $X$ be a Notherian topological space of finte dimension which is Kolmogorov (meaning that for two points $X$ there exists an open subset of $X$ containing one of them but not the other). This ...
3
votes
2answers
76 views

Is every metric space subspace of some connected metric space?

If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected .I'm not sure whether this will work or not, but intuitively I think the ...
0
votes
2answers
33 views

Open connected subset of $ \mathbb R^2 $is path connected [duplicate]

Is open connected subset of $ \mathbb{R^2} $ is path connected?
0
votes
1answer
34 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
1
vote
0answers
15 views

When is a metrizable topological vector space locally bounded?

Consider a topological vector space $E$ with topology $\sigma$. Suppose that $E$ is metrizable, in other words, that there exists a metric $d$ on $E$ that induces the topology $\sigma$. One can then ...
0
votes
2answers
27 views

Klein bottle contains Mobius band

I read the following: "The Klein bottle contains a copy of the Mobius band". I assume this means that there is a subspace of the Klein bottle that is homeomorphic to the Mobius band. How do we obtain ...
0
votes
0answers
10 views

Orientability of Bordered Presentation and its Closure

How can the following claim be true? If $\Pi $ is a bordered presentation, then $\Pi ^c$ is orientable if and only if $\Pi $ is orientable. We know that $\Pi$ must contain border arcs. By ...
2
votes
2answers
37 views

Connected sum of projective plane $\cong$ Klein bottle

How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit homeomorphism, just an ...
1
vote
1answer
29 views

Show compactness of a set given by inequalities

Show that the subset $A=\{(x_1,...,x_n)\in\Bbb R^n |−1≤x_1 ≤x_2 ≤···≤x_n ≤1\}$ is compact. A is contain in an open cover as it is contained in $\Bbb R^n$. Therefore there exists a finite sub cover ...
2
votes
1answer
22 views

$t$-adic topology (on $\mathbb F_p(1/t)$)

Recently I found this interesting discussion about algebraically closed fields of positive characteristic. In the answer marked as the top answer, I read about the $t$-adic topology. The $t$-adic ...
0
votes
0answers
15 views

Performing an Excision on a Topological Surface

Recently I began a book on topology, but the concept of excision on a topological surface isn't clear; perhaps you, collectively, could help elucidate it. Suppose we have an arbitrary topological ...
5
votes
1answer
46 views

Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
0
votes
0answers
22 views

Center of real projective line or Riemann sphere

I have recently encountered the ideas of the real projective line and the Riemann sphere, and it seems to me that in any circle (representing the real projective line) or sphere, the center is a ...
-1
votes
1answer
26 views

Two nonhomeomorphic topological space may be embedded in each other. i need example pls help [on hold]

Find two nonhomeomorphic topological space X and Y such that X imbedded in Y and Y imbedded in X.
0
votes
0answers
33 views

An excerpt from a seminar

It is a statement that a professor made in a seminar which I attended yesterday.He says that the following hold: $1$.If $D$ denotes the closed unit disc then there does not exist a continuous ...
3
votes
2answers
40 views

Characterising the discrete topology with compact subsets [duplicate]

If a set is endowed with the discrete topology then a subset is compact iff it is finite. Is the converse true? That is, given a Hausdorff topological space such that every compact subset is finite, ...
0
votes
1answer
19 views

Two proofs with possibly Baire category theorem about completness.

I'm working with completness right now and I've come across two interesting problems. In my opinion they are worth a little bit attention . a) Let $K$ be closed subset with empty interior on ...
2
votes
1answer
51 views

Decomposition of open sets in $\mathbb{R^d}$

I am trying to prove the following problem. It's an exercise in Stein's Real Analysis text book. Problem: Suppose $\mathbb{R^d}-\{0\}$ is represented as $\mathbb{R_+}\times S^{d-1}$ with ...
1
vote
1answer
25 views

$f:X\to Y \text{ is continuous} \iff f^{-1}(A^*) \subseteq (f^{-1}(A))^*$

Really struggling with exercise 9.10 from Sutherland's "Introduction to Metric and Topological Spaces". Any help would be greatly appreciated. Let $(X,t), (Y,t)$ be topological spaces, and $f: X \to ...
0
votes
1answer
29 views

The closed and bounded sets are compact in the product topology

Let $X=\mathbb{R}^{\aleph_0}$ with the product topology, it is true that all the closed and bounded (in the uniform sense) sets are compact?
1
vote
1answer
39 views

Continuity of function and topology

I have this exercice $E=\{a,b,c,d\}$ with the topology $\tau=\{\emptyset, \{a\},\{a,b\},\{a,b,c\},E\},$ and the space $F=\{x,y,z,w\}$ with the topology $\theta=\{\emptyset.\{y\},\{y,z,w\},F\}$ I ...
1
vote
0answers
12 views

path connectedness of space of almost commuting matrices

Let $R$ be a topological ring which is a domain. Let $n$ be an integer and let $\zeta_n$ be a $n$-th root of unity. Denote by $X$ the set of $m$ by $m$ invertible matrices with coefficients in $R$ ...
2
votes
3answers
41 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...