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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2
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1answer
63 views

Prove or disprove two statements about open functions on metric spaces

Let $f: (X,d) \to (Y,d')$ an open function (not necessarily continuous) between metric spaces. Decide whether the following statements are true or false: 1) If $A \subset X$ doesn't have isolated ...
0
votes
1answer
100 views

Is a continuous map from circle to any topological space necessarily a loop?

I understand that the image of the circle has to be connected and compact. By "loop" I mean the image of the circle is a set X such that there is a surjective continuous function $f: [0, 1] \to X$ ...
1
vote
0answers
25 views

Covering spaces and Automorphisms

I need to find for the groups $G$ a connected degree-4 cover $\hat{B}\rightarrow B$ such that Aut($\hat{B}\rightarrow B$) is isomorphic to $G$ $G \cong 1$ $G \cong \mathbb{Z}_{2}$ $G \cong ...
6
votes
3answers
1k views

“Sum” of positive measure set contains an open interval?

So this homework question is in the context of $\mathbb{R}$ only, and we are using Lebesgue measure. The sum $A+B$ is defined to be $A+B=\{a+b|a\in A,b\in B\}$. The question is: If $m(A),m(B)>0$, ...
0
votes
1answer
61 views

Distinguishing Two Topological Spaces, Each a Union of $\mathbb{S}^1$ and a Line

Currently working on a problem to review for my topology final. Let $\mathbb{S}^1$ be the unit circle $\{(x, y, 0) : x^2 + y^2 = 1\} \subset \mathbb{R}^3$. Let $M = \{(0,0,z) : z \in \mathbb{R}\}$ ...
4
votes
0answers
48 views

Can an irreducible component of a topological space be covered by the other irreducible components?

Let $X$ be a topological space, and write $X=\bigcup X_i$, where the $X_i$ are the irreducible components of $X$. Given any $X_i$, I'd like to find a point $x\in X_i$ such that $x\notin \bigcup_{j\ne ...
2
votes
3answers
124 views

Is Hausdorffness preserved under continuous surjective open mappings?

Is is true that Hausdorffness is preserved under continuous surjective open mappings, I tried to prove it, but I couldn't since even though the images of open sets are open but they need not to be ...
4
votes
2answers
93 views

Show that A=$\{(x_1,…x_n) \in \Bbb R | -1\le x_1\le x_2\le …x_n\le 1\} \subset \Bbb R^n $ is closed.

The full question was: Show that A=$\{(x_1,...x_n) \in \Bbb R | -1\le x_1\le x_2\le ...x_n\le 1\} \subset \Bbb R^n $ is compact, but I was able to show correctly that it is bounded. However my ...
1
vote
3answers
385 views

How to show the intersection of two compact subsets is compact

Let (X,d) be a metric space and A,B $\subset$ X be two compact subsets. Show $A\cap B$ is also compact. I attempted this question by showing the intersection is bounded and closed. But I stated ...
3
votes
1answer
279 views

A surjective ring homomorphism $\phi : C([0,1]) \rightarrow \mathbb{R}$ is evaluation at a point [duplicate]

Let $\phi : C([0,1]) \rightarrow \mathbb{R}$ be a surjective ring homomorphism. How would I prove that $\phi$ is the evaluation map $\phi(f) = f(x)$ for some $x \in [0,1]$? I'm not even sure ...
0
votes
1answer
84 views

Difference between the concepts of graph and trace

I'm a little confused with the definition of graph and trace. If I have a function (or a curve) $f:\mathbb R\to \mathbb R,\ f(t)=t^2$ and I draw the graph we have a parabola since the graph is the ...
3
votes
2answers
1k views

Lower Limit Topology

Show (0,1) is open but not closed in the Lower Limit Topology. I know that [a,b) is open and closed in the lower limit topology, but I am not sure how to prove this one. Thanks for any help.
0
votes
1answer
284 views

Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$

Consider the space $C^1([0,1])$ and the function $d:C([0,1])\times C([0,1]) \to \mathbb R$ defined as $d(f,g)=|f(0)-g(0)|+sup_{x \in [0,1]}|f'(x)-g'(x)|$. Decide whether the metrics $d$ and ...
1
vote
1answer
90 views

for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous.

Let $X$ and $Y$ be two topological spaces. Prove that for any $A\subseteq X$, $f(\overline{A})\subseteq\overline{f(A)}$ , if and only if $f: X \to Y$ is continuous. I am stuck on the converse. ...
0
votes
1answer
85 views

Universal covering Spaces Drawings

I just have trouble drawing universal covers, how can I draw the universal covers of the following spaces: $X$ is the union of a circle with a projective plane $\mathbb{P}^2$ identified along a ...
1
vote
1answer
126 views

Proving a subset $A$ of a metric space $(X,d)$ is open

Let $(X,d)$ be a metric space and let $A \subset (X,d)$. Prove that $A$ is open iff for every sequence $\{a_n\}_{n \in \mathbb N}$ such that $lim_{n \to \infty} a_n \in A$, there exists $n_0 : a_n \in ...
0
votes
1answer
69 views

Union of closed normal subspaces is normal

Let $X=F_{1} \cup F_{2}$ be a space such that $F_{1}$ and $F_{2}$ are closed normal spaces(as subspace to $X$) I need to prove that $X$ is also normal My attempt goes as follow: From normality of ...
0
votes
1answer
29 views

Existence of measure under inverse transformation

Suppose there is nonempty compact metric spaces $X$, $Y$ and a continuous surjective transformation $T : X \to Y$. For given finite measure $\nu$ on $(Y,\mathcal{B})$, is a measure $\mu$ on $(X, ...
3
votes
1answer
360 views

Besicovitch Covering Lemma

We just finished our unit on covering lemma's in my analysis class and my professor proved both the Vitali and Besicovitch covering lemma's (for finite and infinite coverings) using balls. He ...
2
votes
3answers
166 views

Why is ($\mathbb R$,usual) not homeomorphic to ($\mathbb R$,discrete)?

Why is ($\mathbb R$,usual) not homeomorphic to ($\mathbb R$,discrete)? ($\mathbb R$,discrete) means $d(x,y) =1$ for any $x\neq y$ and $d(x,y) =0$ for all $x=y$, both $x$ and $y$ are in $\mathbb R$. ...
0
votes
1answer
49 views

Determining whether this is a group action

I'm having trouble with an exercise we were given. I have to determine for which values $a,b\in\mathbb{R}$ $$n\cdot t=\phi_n(t)=2^nt+a^n+b$$ defines a group action of the group $(\mathbb{Z},+)$ on ...
2
votes
2answers
106 views

basic question involving topology and the Hausdorff distance

Let $\Omega_1 \supset \Omega_2 \supset ...$ a sequence of nonempty , open, bounded and convex sets. Define $\Omega = int \Bigl( \overline{\displaystyle\bigcap_{k=1}^{\infty} \Omega_k } \ \Bigl) $ and ...
1
vote
2answers
61 views

Is every regular star compact metaLindelof space compact?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. Star ...
4
votes
1answer
93 views

Connected 0-measure set does not contain line segments

Is there a set which is a subset of $\mathbb{R}^n,n>1$, is connected, has a positive area (Lebesgue measure) and does not contain any single line segment?
1
vote
2answers
45 views

how can i prove that the sorgenfrey line is not sigma compact?

Can someone give me a hint to prove that the sorgenfrey line is not sigma compact? thanks in advance
3
votes
0answers
77 views

Relative continuity of a function

I've come across the following generalization of continuity in a rather surprising place. Let $X$, $Y$, and $Z$ be topological spaces and $f:X\to Y$ and $g:Y\to Z$ be functions. Say $f$ is continuous ...
10
votes
3answers
768 views

Functions from the space of subsets of a set to the space of topologies

Let $X$ be a set and $\mathcal P \left({X}\right)$ be the set of subsets ordered with inclusion. Let $\mathcal{T}(X)$ be the set of all topologies on $X$ ordered with set inclusion. Let ...
1
vote
1answer
64 views

Prove that set is arcwise-connected. Function $\mathbb{R}^3 \to \mathbb{R}^2$ with differential of rank $2$.

Let $f:\mathbb{R}^3 \to \mathbb{R}^2$ and assume that $0$ is regular value of $f$ (i.e. the differential of $f$ has rank $2$ at each point of $f^{-1}(0)$). Prove that $\mathbb{R}^3 \setminus ...
4
votes
1answer
120 views

The sheaf $\mathfrak{S}$ of germs of analytic functions over $D$ is a topological group (Ahlfors)

In Ahlfors' complex analysis text, page 286 he gives the following definition: Definition 1. A sheaf over $D$ is a topological space $\mathfrak S$ and a mapping $\pi:\mathfrak S \to D$ with the ...
2
votes
1answer
60 views

Can the ultrafilters in the poset of open subsets be made into a topological space?

Let $X$ denote a topological space and $O$ denote its poset of open subsets. Intuitively, $O$'s ultrafilters are kind of like generalized points of $X$. Is there a way to make these ultrafilters into ...
1
vote
2answers
109 views

Error in Engelking?

I've been trying to do some exercises in Engelking's General Topology text, and there's one that's causing me problems. I hope that someone here can clarify this for me. The exercise is (slightly ...
2
votes
2answers
114 views

Continuous bijection whose inverse is not continuous at uncountably many points

I am interested in understanding to what extent continuous bijections fail to be homeomorphisms. For example, suppose $X, Y$ are metric spaces and $f: X\to Y$ is a continuous bijection. Is it possible ...
6
votes
2answers
90 views

Has the idea of generalizing the codomain of a metric been seriously considered?

The long line is much longer than $\mathbb{R}$, and indeed many chains have this property. Thus, since metrics are usually assumed to be real-valued, this can be understood as an assumption that ...
0
votes
0answers
52 views

Quasicompact? Why the distinction?

What is the reason that some topologists use quasicompact? Why is the distinction made? quasi means "not really", so why use this terminology?
0
votes
1answer
39 views

If $X=\{a,b,c,d\}$ with a) the discrete topology and b)indiscrete topology, are they normal, regular, or both?

So if $X=\{a,b,c,d\}$ with $\mathcal{D}$ the discrete topology is is normal, regular or both? I have its neither normal or regular since the discrete topology is all subsets of X that are open and ...
2
votes
1answer
75 views

question on closure and interior of a set.

In any topological space, is there an open set $G$, and two sets $A$, $B$ such that $\operatorname{cl}(G)$ contained in both $\operatorname{cl}(\operatorname{int} A)$ and ...
0
votes
0answers
56 views

Topology on function Spaces

I was wondering, what are the different topologies that are usually given to function spaces? Sorry if this is a broad question. For example, I know that compact-open (under nice assumptions) leads ...
0
votes
1answer
39 views

the discrete family of closed sets cover

When I study on the proof of every paracompact space is collectionwise normal, I have problem. Let $X$ be a paracompact space and $\{F_s\}_s\in S$ be a discrete family of closed subsets of $X$. Then ...
1
vote
1answer
54 views

In the proof the proving duality of topological vector space

I find hard to understand the proof for this theorem. Hope that some one can help me to clarify this. Thanks Suppose $X$ is a vector space and $X'$ is a separating space of linear functionals on ...
1
vote
1answer
66 views

Closed sets, boundary, topology.

Let A be a closed subset of the real numbers. It is always possible to find a subset B of the real numbers such that A is equal to the boundary of B? Prove if true, find a counterexample if not. I ...
3
votes
1answer
55 views

Homotopy vs Conservative

I learned about conservative fields in multivariable calculus. I'm always curious about finding other or more fundamental methods of describing a concept (or concepts) in math, and better ...
2
votes
1answer
61 views

Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1} $$are continuous. Is the single requirement that ...
4
votes
2answers
267 views

Show that the cone of the open interval (0, 1) can not be embedded in any Euclidean space

I've been trying to tackle this problem for some while now, but don't know how to start correctly. I know that the cone on $(0,1)$ is given by $$\text{Cone}((0,1)) = (0,1) \times ...
2
votes
2answers
126 views

On eventually constant sequences

It is of course true that in a discrete space a sequence converges iff it's eventually constant. Is the converse true, i.e., if the only convergent sequences in a space are eventually constant, is the ...
0
votes
1answer
40 views

$V=L^2(\Omega,Z)$ is path connected

Let $V=L^2(\Omega,Z)$. Prove that V is path connected by paths of class 1/2 Holder. I would appreciate it if anyone could give me a suggestion. Thank you in advance.
4
votes
1answer
123 views

Outline of a proof that $\mathbb{R}^2 - A$ where A is countable is path-connected

Let $A$ be a countable subset of $\mathbb{R}^2$. Show that $\mathbb{R}^2-A$ is path connected. These are my steps: Let $x$ and $y$ be arbitrary points of $R^2$ Let $f^r:[0,1] \to \mathbb{R}^2$ be ...
1
vote
1answer
61 views

Fundamental Group of $S^3$ \ A , where A is a figure 8.

I'm trying to figure out the fundamental group of a topological space X obtained from $\mathcal{S}^3$ by removing from it a "figure 8", i.e. the set $K=\{(x_1,x_2,0,0)\in\mathbb{R}^4 | x_1^2 + ...
0
votes
1answer
170 views

Proving $\mathbb R^2 \setminus \mathbb Q^2$ is connected [duplicate]

I am trying to prove that the set $\mathbb R^2 \setminus \mathbb Q^2$ is connected. I don't know if the following is true: could it be that it is also path connected? If that is the case, maybe it's ...
2
votes
2answers
163 views

Proving that this space is path connected.

If $X$ is a topological space, then I have to show that $(X \times D^1)/$~ is path connected (where $(x_1, y_1)$ ~ $(x_2, y_2)$ $\iff$ $y_1=y_2=1$ or $y_1=y_2=-1$ or $(x_1, y_1) = (x_2, y_2)$ ). Let ...
1
vote
1answer
317 views

Is path-connected and locally path-connected equivalent to simply connected?

Is path-connected and locally path-connected equivalent to simply connected? I am more confident with the part that simply connected implies path-connected and locally path-connected. And the other ...