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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3
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27 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
1
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2answers
28 views

continuous map of connected set is connected, example: Proving the connectedness of this set.

I thought I would try to use this to prove connectedness in this set if possible: $$\{(x,y)\mid 1<x^2+y^2<4\}$$ $f(x,y)=x^2+y^2$ So since $(1,4)$ is connected in $\mathbb R$ so it this set, as ...
3
votes
0answers
21 views

What are the modes of vibration of a genus-2 surface?

So it's spherical harmonics for a sphere. The vibrations of a torus presumably are just ordinary string harmonics around each loop. But what are the harmonics on a genus-2 surface (a donut with 2 ...
-2
votes
1answer
46 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
0
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0answers
53 views

Trying to prove that a continuous function $\,f: \mathbb{D}^2 \rightarrow \mathbb{R}^2$ has a zero using the fundamental group?

Prove: For a continuous $\;f: \mathbb{D}^2 \rightarrow \mathbb{R}^2,\;$ $\,f\left(\omega,0\right) \notin f\left(S^1\right),\,$ given a non constant function $\left(f_{\mid S^1}\right)_{\ast} : ...
2
votes
2answers
67 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
1
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1answer
49 views

Discontinuous $ f : \mathbb R^2 \to \mathbb R$ with unusual topology on $ \mathbb R$

With the usual topology on the reals $\mathbb R$ , let $D$ be the family of dense open sets and let $T=D \cup \{ \phi \}$. Let $S$ be the set $R$ with the topology $T$ on it. Show that the function ...
6
votes
1answer
205 views

What type of surface is it?

The picture shows Sphere, Torus, Klein Bottle and Projective Plane, respectively: What about the following one? Is it also Projective Plane? : PS inside triangles and color in shapes are ...
0
votes
1answer
31 views

If $X, Y$ are topological spaces, does $A \times B$ closed in $X \times Y$ imply $A$ closed in $X$ and $B$ closed in Y?

Suppose $X, Y$ are topological spaces, and $A\subset X$, $B\subset Y$. Does $A \times B$ being closed in the product space $X \times Y$ (with the product topology) imply that $A$ is closed in $X$ and ...
1
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0answers
21 views

A question uniform convergence

Let $X$ be a compact Hausdorff space, $a$ a continuous real-valued function on $X$, and for $t\in\mathbb{R}$ let $f_t(x)=\exp(ia(x))$ such that the function $t\mapsto f_t$ is continuous (where we use ...
1
vote
1answer
35 views

If a compact subset is contained in an open subset in $\mathbb{R}^n$, is a small cylinder of this compact subset also contained in the open set?

Let $O\subseteq \mathbb{R}^n$ be an open set, $K\subseteq \mathbb{R}^{n-1}$ a compact set and $a\in \mathbb{R}$, such that $$\{a\}\times K\subseteq O$$ holds. Does there exist an $\epsilon>0$, ...
1
vote
1answer
35 views

Example of a separable, locally-compact metric space which is not $\sigma$-compact

I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact. At first I thought I could show that if a metric space is separable and locally-compact, then ...
1
vote
2answers
57 views

How does convergence imply continuity?

I'm trying to develop some background understanding to eventually prove the following: ....................... Let $M$ and $N$ be metric spaces and let $f : M \rightarrow N$ be a map. Show that $f$ is ...
0
votes
1answer
42 views

Prove $S^1\times I\to D^2,(s,t)\mapsto ts$ is an identification.

First of all, every map here is continuous. I'm trying to prove the following: If every $S^1\to X$ is homotopic to a constant map $S^1\to X$ then every map $S^1\to X$ extends to a map $D^2\to X$ ...
4
votes
2answers
132 views

Prove that closed subsets of a compact set is compact. What's wrong with this proof?

I understand other methods of achieving the result, but this was my first try. I'm not sure where my mistake is, if any. And yes, I realize that using the fact that $B$ is closed would help. For a ...
5
votes
1answer
94 views

Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$?

Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if ...
1
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2answers
65 views

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial.

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ is continuous and has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial. I'm not sure how the fact that there exists an ...
3
votes
1answer
84 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...
1
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0answers
31 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
1
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4answers
75 views

Question on compact sets in $\mathbb{R}^2$

Let $\Omega$ be an open set in $\mathbb{R}^2$, $K \subseteq \Omega$ and K be compact. Prove that there exists $r>0$ such that $E=\bigcup_{z \in K} \bar{D}(z,r)$ is a compact subset of $\Omega$, ...
0
votes
1answer
39 views

Is my proof that the product of covering spaces is a covering space correct?

Let $p_1:\tilde X_1 \rightarrow X_1$ and $p_2:\tilde X_2 \rightarrow X_2$ be two covering spaces. Prove: $p = p_1 \times p_2:\tilde X_1 \times \tilde X_2 \rightarrow X_1 \times X_2$ is a covering ...
1
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1answer
55 views

What is the relation between $\operatorname{int}f(S) $ and $ f(\operatorname{int}S)$ & $f(\overline S)$ and $ \overline {f(S)} $

$\newcommand{\int}{\operatorname{int}}$Let $ f: \mathbb R→ \mathbb R $ be a continuous function and let $S$ be a non-empty proper subset of $\mathbb R$ . Which one of the following statements is ...
0
votes
1answer
31 views

Intersection of arbitrary union of compact subsets.

My textbooks asks to prove that arbitrary intersection of compact subsets in hausdorff space is again compact. I've kinda found the counterexample $\bigcap_{1\leq x<2} [x,3]=(2,3]$, and can't find ...
1
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1answer
40 views

oThe cofinite topology is the infimum of the hausdorff topologies in a set $X$

Let $X$ be any set and $\tau_c$ the cofinite topology in $X$ and $T$ be the set of hausdorff topologies on $X$. Prove that $\tau_c=\inf T$ where the relation is inclusion in the set of all topologies ...
2
votes
0answers
49 views

product of spaces is a manifold. Are the spaces?

Suppose that $X$ and $Y$ are topological spaces and that $X\times Y$ is a topological manifold. It seems that we can't conclude that $X$ or $Y$ are manifolds themselves (this question). EDIT :Are ...
4
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0answers
55 views

Simpler version of dogbone space construction

In "The cartesian product of a certain nonmanifold and a line is $E^4$" (R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) Bing constructs a nonmanifold, $B$, such that $B\times \Bbb ...
2
votes
1answer
95 views

Let $f:M\to N$ be continuous, then $f(U)\subset V$.

Let $f:M\to N$ be continuous, with $M$ and $N$ metric spaces. Given an arbitrary subset $X \subset M$ and an open set $V \subset N$, with $f(X) \subset V$, prove that there exists an open subset $U$, ...
8
votes
0answers
66 views

Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
2
votes
2answers
99 views

An intitutive solution to problems relating to closed sets in topology

The question given in my homework problem is, Let $ \{A_{\alpha}\}_{\alpha \in \Lambda} $ be a family of closed subsets in an arbitary topological space $X$ . Assume that for each $x$ there exists an ...
0
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0answers
32 views

Topology Question in Munkres Analysis on Manifolds

Let X be a metric space with metric d. Let Y $\subset$ X. Give an example where A is open in Y but not open in X. Give an example where A is closed in Y but not closed in X. I'm stuck on question ...
2
votes
1answer
66 views

The sphere $S^2$ is not contractible

I heard that in topology the sphere $S^2$ cannot be continuously deformed to a point, i.e. $S^2$ is not contractible. Sorry for my ignorance, but I really don't get it. Can't we just push all the ...
2
votes
2answers
76 views

In algebraic topology, for a function $f$ what does $f _\ast$ mean?

In algebraic topology, for a function $f$ what does $f_{ \ast}$ mean? I'm solving some exercises and this is something that's appearing, often relating to homotopic functions, and I'm not sure what ...
2
votes
2answers
43 views

Proof of Urysohn's lemma from Kelley's book.

I am (self)-studying general topology from the Kelley's book "General Topology", and there is a proof that I don't manage to understand. I am speaking about the Urysohn's lemma: If $A$ and $B$ are ...
2
votes
1answer
47 views

Help me understand this passage from “General Topology” by J. Kelley

I'm having trouble understanding what is meant with the following passage: ... functions s.t. $S(m,n)$ is defined whenever $m$ belongs to a directed set $D$, and $n$ belongs to a directed set ...
-1
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0answers
23 views

What does a homotopic lift mean when talking about homotopic functions? [closed]

I misunderstood the problem I was working on, this should be closed.
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0answers
31 views

If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$. [on hold]

This is theorem 36.2 in Topology - James Munkres. I don't know how the piecewise function $h_i$ has the same value for when the domains intersect. Because $A_i \subset U_i,$ some $x \in (X - A_i) \cap ...
3
votes
1answer
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
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votes
3answers
95 views

Is this true: Every open set $A$ contains a neighborhood whose closure is a subset of $A$. [closed]

This seems a very easy fact. But I don't know how to prove it. Can anybody help me? Thanks!
1
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2answers
56 views

Critique my proof from munkres product topologies?

I've been going through Munkres' book on topology on my own, and I just struggled through the proof of 10d) from chapter 2 section 19. I've never had a chance to show one of my proofs to anyone, so I ...
0
votes
1answer
27 views

Conjecture about regular Borel measures and dense sets with no interior

Suppose that $(X,\tau)$ is a topological space and let $\mathscr B$ denote the Borel $\sigma$-algebra on it. Moreover, let $\mu:\mathscr B\to[0,\infty]$ be a regular Borel measure, that is, ...
5
votes
1answer
120 views

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
0
votes
1answer
30 views

Simlpe Loops in Topological Graph

Given a set of points in 3D space, and a set of links between them which form a connected graph - is there a general strategy for extracting all simple loops from such an object? I refer to simple ...
6
votes
1answer
32 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
0
votes
3answers
91 views

Definition of compactness unnecessarily verbose?

The definition of a compact set is given as a set, $X$, for which all open covers have a finite subcover. This seems unnecessarily verbose to me. Wouldn't it be sufficient to simply say that $X$ has ...
0
votes
1answer
21 views

Is a limit point compact subset of a Hausdorff space necessarily closed?

This is Exercise 3 c) from Section 28 of Munkres - Topology. I had thought the answer was no initially, but the example I came up with was not Hausdorff. At this point, I'm just not sure. Here's ...
2
votes
1answer
54 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
5
votes
1answer
57 views

Continuity of a “minimal distance” projection $f:(X,d) \to (K, d_{|K})$ for a compact $K \subset X$. (Hint preferred)

Let $(X,d)$ be a metric space and $K$ be a compact subset of $X$. Show that for every $x \in X$ there exists $k_x \in K$ such that $$d(x,K)=d(x,k_x)$$ Suppose that for every $x\in X$, there exists ...
0
votes
0answers
15 views

Are some some particular subspaces of cadlag functions Polish?

Consider the space $D := D((0, \infty), \mathbb{N})$ of cadlag functions $f : (0, \infty) \to \mathbb{N}$ equipped with the Skorokhod $M_1$-topology. Then $D$ is Polish. Question 1: I want to check ...
0
votes
1answer
28 views

Proof that a homeomorphic image of a non-borel set is non-borel

This question seeks to expand the proof given in the answer to this question. I am weak in topology, and am wondering if someone can provide a proof of why a homeomorphic image of a non-borel set is ...
-3
votes
4answers
73 views

A compact open set

Is there an open set which is compact ? I would say that $\emptyset$ is an open set compact because it's bounded and closed too. Is it correct ?