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
27 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 = ...
1
vote
1answer
25 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
3answers
36 views

Normal matrices connected?

Is the set of all normal matrices connected in $M_n(\mathbb{R})$, where the metric is the usual metric of $\mathbb{R}^{n^2}$? ($A$ is normal iff $AA^{t}=A^{t}A$.)
5
votes
1answer
113 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 ...
3
votes
0answers
29 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
15 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
1answer
33 views

Möbius band parameterizaton: Showing injective

So, I'm trying to show that the parameteization function from $\mathbb R^2$ to $\mathbb R^3$ given in the wikipedia page http://en.wikipedia.org/wiki/Mobius_band#Geometry_and_topology is injective on ...
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
32 views

Limit points of a set

If $A=\{\frac{2}{m}+\frac{3}{n}:m,n\in \mathbb N\}$, then what is the derived set of $A$ in $\mathbb R$? Definitely $0$ is a limit point of $A$. I think for all $n\in \mathbb N$, all numbers of the ...
-1
votes
2answers
19 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 ...
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
0answers
51 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 ...
0
votes
1answer
37 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?
3
votes
0answers
36 views
+50

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
0
votes
2answers
19 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 ...
0
votes
3answers
37 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
1answer
32 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 ...
3
votes
2answers
51 views

Existence of strictly positive probability measures

Let $X$ be a Hausdorff space (or let's even assume it is metrizable). A strictly positive measure on $X$ is the that gives positive measure to any non-empty open subset of $X$. Under which condition ...
1
vote
0answers
91 views
+50

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
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
18 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
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
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
47 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 ...
2
votes
3answers
98 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of $\mathbb{R}$ and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
3
votes
2answers
75 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 ...
3
votes
0answers
52 views
+500

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
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 ...
0
votes
0answers
7 views
+300

Distributivity of group topologies on a Prüfer group

Let $p$ be a prime number and $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Z_{p^\infty}$ such that $(\Bbb Z_{p^\infty},\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ ...
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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
41 views

continuous poset w.r.t. Scott topology

I am learning continuous poset by myself. I have conclusion as follows: If $P$ is a continuous poset w.r.t. Scott topology then there is $x\in P$ s.t. for any $y\in P$ and for any open sets $U_x$ and ...
5
votes
2answers
152 views

Open subspaces of locally compact Hausdorff spaces are locally compact

Let $X$ be locally compact and Hausdorff. I want to prove the following: If $Y\subset X$ is open then $Y$ is locally compact. I have proved that closed subsets of $X$ are locally compact, but ...
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?
1
vote
0answers
14 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 ...
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
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 ...
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 ...
1
vote
1answer
20 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 ...
3
votes
1answer
63 views

Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
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
0answers
24 views

Is $S(\mathbb{R}^{d})$ dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$?

Let $S(\mathbb{R}^{d})$ denote the class of Schwartz functions in $\mathbb{R}^{d}$. Is it true that $S(\mathbb{R}^{d})$ is dense in $L^{1}_{\textrm{loc}}(\mathbb{R}^{d})$, the locally integrable ...