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
137 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
99 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
vote
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
88 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
vote
0answers
33 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
vote
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
40 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
vote
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 ...
2
votes
1answer
52 views

The 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
votes
0answers
56 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
97 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 $U$, ...
4
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
100 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
votes
0answers
34 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
78 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 ...
-2
votes
0answers
31 views

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

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
128 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 ...
-1
votes
3answers
96 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
vote
2answers
57 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
28 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
123 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
94 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
56 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
58 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
16 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 ?
3
votes
1answer
56 views

If $U\subseteq V \subseteq \mathbb{R}^{2}$, then $\partial U \cap V =\partial\left(V\setminus U\right)\cap V$?

Assume $U\subseteq V \subseteq \mathbb{R}^{2}$, both open, is then $\partial U \cap V =\partial\left(V\setminus U\right)\cap V$? Edit: What if $U, V$ also are simply connected?
2
votes
1answer
90 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
1
vote
1answer
53 views

Countable choice and totally bounded metric spaces

Can we prove that the following statement is equivalent to the axiom of countable choice (CC)? If every sequence in a metric space $X$ has a Cauchy subsequence, then $X$ is totally bounded. ...
0
votes
0answers
47 views

Open Set in the Cartesian plane.

I'm trying to prove that the following set is an open set in $\mathbb{R}^2:$ $$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$ with respect to norm $||x||_{1},||x||_{2},||x||_{\infty}.$ ...
0
votes
1answer
19 views

Is the condition “the inverse image of a closed base set is closed?” sufficient for continuity?

Let's say you have a function $f:X \to Y$, where $X$ and $Y$ have topologies. The set $C$ forms a closed base for $Y$. If for every $c \in C$, $f^{-1}(c)$ is closed in $X$, is $f$ continuous? If the ...
1
vote
1answer
56 views

The empty set is a neighborhood?

The following axioms of a Topological space is from Wikipedia: Neighbourhoods definition This axiomatization is due to Felix Hausdorff. Let $X$ be a set; the elements of $X$ are usually ...
5
votes
2answers
61 views

Prove $\{(x,y): x>0\}$ is connected

As an introduction to multivariable calculus, I'm given a small introduction to some topological terminology and definitions. As the title says, I have to prove that $\{(x,y): x>0\}$ is connected. ...
0
votes
0answers
18 views

Deforming path of integration from the real line to the boundary of a open subset of the upper half complex plane.

Denoted the upper half of the complex plane by $\mathbb{C}^{+}=\{z\in\mathbb{C}:\text{Im }z>0\}$. Let the open, unbounded set $A\subseteq\mathbb{C}^{+}$ have a boundary $\partial A$ such that the ...
2
votes
1answer
72 views

Topology(meaning) [closed]

When we define Topology we say that a topology on a set(let's say X) is a collection of subsets of X having certain 3 properties. Now, here what do we actually mean by saying "topology on a set". What ...
2
votes
1answer
36 views

Image of isometric immersion

Let $M$ a metric space with the following property: For all isometric immersion $f:M\to N$, the image of $f(M)$ is a open set in $N$. Prove that $M$ is empty set. A function $f:M\to N$, is called ...
0
votes
2answers
45 views

Show that if A is bounded above, then it contains its supremum…

Please check my answer. ..........Question......... Suppose $A \subseteq \mathbb{R}$ is closed and nonempty. Show that if $A$ is bounded above, then it contains its supremum, and if it is bounded ...
2
votes
2answers
20 views

Locally compact Hausdorff space and indicators

This is exercise 6 from Tao's notes on locally compact Hausdorff spaces. Let $X$ be such a space and assume $K \subset U$ where $K$ compact and $U$ open. We want to find a function $f:X \to \mathbb ...
3
votes
1answer
56 views

Is $f(x,y)=ax^2+by^2, \ a,b \in \mathbb R $ a bijection between $\mathbb R^2 \to \mathbb R$? Bijections of topologies

Is $f(x,y)=ax^2+by^2$ a bijection between $\mathbb R^2 \to \mathbb R$ ? How about $f(x,y,z)=\frac{x^2}{a^2} + \frac{y^2}{b^2}+ \frac{z^2}{c^2}? ( \mathbb R^3 \to \mathbb R )$ What confuses me now ...
1
vote
2answers
52 views

If $A\subseteq B\subseteq\mathbb{R}^{2}$, $A$ open, $B$ closed, then $\overline{A}\subseteq B$?

If $A\subseteq B\subseteq\mathbb{R}^{2}$, $A$ open, $B$ closed, then $\overline{A}\subseteq B$? That is, $\partial A \subseteq B$?
1
vote
0answers
37 views

Are these subsets open, closed, both or neither (revised)?

This is a follow up to Are these subsets open, closed, both or neither? Please let me know if my answers are correct, and If my reasoning is accurate and complete. Below are my corrections: ...