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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7
votes
1answer
169 views

Is there a $C^1$ curve dense in the plane?

Is there a curve $\gamma : \mathbb{R} \to \mathbb{R}^2$ injective and $\mathcal{C}^1$ whose range is dense in $\mathbb{R}^2$?
4
votes
1answer
116 views

Algebraic $n$-torus and topological $n$-torus

Working with the field $\mathbb C$, one can find two different objects called "torus": - Algebraic $n$-Torus that is $(\mathbb C^\ast)^n$ - Topological $n$-Torus defined (for example) as the ...
2
votes
1answer
107 views

Subspace of $\ell_\infty$ that is not separable

I need to prove that $$L = \left\{ (x_i)\in \ell_\infty : \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n x_i = 0\right\}$$ is a subspace in $l_\infty$ (done that), is closed in $l_\infty$ (done ...
4
votes
1answer
214 views

Isometric Isomorphism from $c_0$ to $C[0,1]$

I need to prove that there is an isomorphic isometry from $c_0$ to some subspace of $C[0,1]$. Researching a bit, it looks like it follows from Banach-Mazur theorem, but we haven't studied it, at least ...
1
vote
1answer
99 views

Determining if these sets are compact?

Which of the following sets are compact? (a) {$(x,y)\in\mathbb{R}:2x^2-y^2\leq1$} (b) {$x\in\mathbb{R}^n:2\leq||x||\leq4$} (c) {$(e^{-x}\cos{x}, e^{-x}\sin{x}):x\geq0$} $\cup$ {$(x,0):0\leq ...
3
votes
2answers
496 views

Going from a fundamental system of neighborhoods to a topology and vice versa

Given a topological space $(X,\tau)$ and a point $x\in X$ we can define a fundamental system of neighborhoods of $x$ (or perhaps a neighborhood base at $x$), say $\mathscr{N}(x)\subseteq2^X$, by every ...
2
votes
5answers
196 views

How to show that the vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$?

The vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$. How to show this?
0
votes
1answer
49 views

Some questions from Arhangel'skii-Buzyakova

The proposition 2.6 of On linearly Lindelöf and strongly discretely Lindelöf spaces_ by Arhangel'skii and Buzyakova: Let $X$ be a linearly lindelof Tychonoff space of countable tightness such hat ...
8
votes
1answer
230 views

Why not just study the consequences of Hausdorff axiom? What do statements like, “The arbitrary union of open sets is open,” gain us?

Define that a pair $(X,\tau)$ where $\tau \subseteq \mathcal{P}(X)$, is a Hausdorff space if for all distinct $a,b \in X$ there exist $A,B \in \tau$ such that $$a \in A, \;b \in B, \;A \cap B = ...
3
votes
2answers
89 views

A question on a dense subspace

Suppose $Z$ is a topological space; and $X$ is dense in $Z$. Then do we have $W(X)= W(Z)$? Where $W(X)$, $W(Z)$ denote the weight of the $X$ and $Z$ respectively. What I've tried: On one hand, ...
5
votes
1answer
78 views

Every first countable space is a moscow space.

First countable space $X$ is an example of moscow spaces. Let $U$ is an open subset of $X$ and $x\in \overline{U}$. If $\overline{U}$ is open or even a nbhood of $x$ this proposition is immediately ...
1
vote
1answer
75 views

Definition of a perfect point set

I was reading an older paper and at one point they mention something called a "perfect point set" - it's in the lemma on the second page. I ran into trouble trying to find an actual definition. Is it ...
1
vote
3answers
228 views

Metrizability is a topological property?

How could I show that metrizability is a topological property? Well, this means that if I have a set X that is metrizable and a homeomorphic function f from X to Y, then I need to show that Y is ...
3
votes
2answers
63 views

Proving that certain subspace of $\ell_1$ is non closed

I need to prove that $$L= \left\{(x_i) \in\ell_1 : \sum_{i=1}^\infty ix_i= 0\right\}$$ is non-closed in $\ell_1$. I can't really think of sequences of sequences that are in this subspace, much less ...
5
votes
0answers
108 views

Homological definition of orientation at a boundary point?

For a topological manifold $M^m$, an orientation at a point $x \in M$ can be defined as a choice of generator for $H_m (M, M-x)$. For a topological manifold with boundary this definition still makes ...
1
vote
1answer
121 views

Totally bounded set, $\varepsilon$-nets

I'm working through Martin Schechter's "Principles of Functional Analysis" (2nd ed.) and a problem concerning totally bounded subsets and $\varepsilon$-nets of normed linear vector spaces on page 96 ...
11
votes
1answer
277 views

Space of homeomorphisms Homeo$(S^1)$ of $S^1$ deformation retracts onto $O(2)$

How can we prove that the space of homeomorphisms Homeo$(S^1)$ of $S^1$ (strong) deformation retracts onto the orthogonal group $O(2)$? I know that this result is proved by Hellmuth Kneser in his ...
19
votes
5answers
566 views

Is the closure of a Hausdorff space, Hausdorff?

$(X,\mathcal T)$ is a topological space which has a dense Hausdorff subspace. Is $X$ Hausdorff?
4
votes
2answers
1k views

Closed Bounded but not compact Subset of a Normed Vector Space

Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is $||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. Prove that $B'(0,1) = \{x \in l^\infty ...
0
votes
1answer
40 views

Understanding the relation between countably paracompact and monotonically normal

Does monotonically normal imply countably paracompact? Thanks ahead:)
3
votes
0answers
187 views

A question from Arhangel'skii-Buzyakova

Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.) ...
5
votes
1answer
181 views

Is the configuration space a manifold? a CW complex?

The ordered configuration space of $n$ points in a topological space $X$ is defined as $F(X,n)=\{(x_1,\ldots,x_n)\in X^{n} | x_i\neq x_j \mbox{ for } i\neq j\}$ and the unordered configuration space ...
0
votes
1answer
117 views

How to prove inverse metric is a metric in Y

I just need to know if my answer is right based on the following question: Let $(X,d)$ be a metric space, and let $f:X\to Y$ be a homeomorphism of $X$ onto a topological space $Y$. Define a metric ...
1
vote
1answer
31 views

Anti-symmetric property of embedding in topological spaces.

$(X,\mathcal T)$ and $(Y,\mathcal S)$ are topological spaces. $X$ can be embedded homeomorphically in $Y$ and $Y$ can be imbedded homeomorphically in $X$. Are $X$ and $Y$ homeomorphic? How about ...
0
votes
1answer
726 views

what does 'arbitrary' mean?

Can arbitrary union of open intervals be written as countable union of open intervals? Here, I don't even understand what do they mean by arbitrary. Help me please.
2
votes
0answers
39 views

Topological inequivalent manifolds obtaining by removing a surface from a manifold

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
0
votes
2answers
161 views

How would I show that if $S,T\subset M$ and $S\subset T$ then $cl(S)\subset cl(T)$?

I cant really think of anything to say other then to say that since $S\subset T$ then that's basically why cl$(S)$$\subset$ cl$(T)$. But that just seems too easy and it's probably not right. So I ...
2
votes
2answers
184 views

prove Taylor of $R(a)$ converges $R$ but its sum equals $R(a)$ for $a$ in interval.Which interval? Pls I'm glad to give an idea or hint?:

$T(a)=\begin{cases} 0 & a\leq 0\\ e^{-1/a} & a>0 \end{cases}$ $Z(a)=\begin{cases}0 & a\geq 1\\ (1+a) e^{-1/(1-a)} & a<1 \end{cases}$ $p=\int_{-1}^{1}Z(x)dx$ ...
4
votes
1answer
97 views

Prove $\operatorname{dist}(\overline{A},\overline{B}) = \operatorname{dist}(A, B)$

This is the last question on the exercise sheet and I am having real trouble formalizing my intuitions. It should be obvious. Since the closure of a set is the set of all points in the universe with ...
4
votes
2answers
92 views

How does one show that there exists some $z \in X$ such that $f(z) = z$ under certain circumstances?

In a previous exercise, one was asked to show that the sequence $(x_n)_{n > 0}$ in $X$ (with $(X,d)$ a non-empty, complete metric space) in which we have $d(x_n,x_{n+1}) \leq \theta d(x_{n-1} , x_n ...
1
vote
2answers
64 views

Elementary topology problem.

Let $ ((Y_{\alpha},\tau_{\alpha}) \mid \alpha \in J) $ be a $ J $-indexed family of topological spaces and $ X $ any non-empty set. Let $ (f_{\alpha} \mid \alpha \in J) $ be a $ J $-indexed family of ...
4
votes
2answers
213 views

Are these subsets of $\mathbb{R}$ homeomorphic?

Consider the following subspaces of $\mathbb{R}$ with the usual topology: $$X = (0, 1) \cup \{2\} \cup (3, 4) \cup \{5\} \cup \cdots \cup (3n, 3n + 1) \cup \{3n + 2\} \cup\cdots$$ $$Y = (0, 1] \cup ...
2
votes
1answer
233 views

Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected

Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected. (Hint:Show that given $x_0 \in U$, the set of points can be joined to $x_0$ by a path in $U$ is both open ...
2
votes
0answers
68 views

Prove that you can't connect both pairs of opposite sides of a square without the two paths intersected.

Formally, let $$D=[-1,1;-1,1]\subset\mathbb{R}^2,$$ and let $f,g:[0,1]\to D$ be two continuous functions, such that $f(0)=(-1,0)$, $f(1)=(1,0)$, $g(0)=(0,-1)$, $g(1)=(0,1)$. Prove that ...
3
votes
3answers
398 views

$f$ continuous, $f: X \to Y$, $Y$ compact Hausdorff. Is $f(X)$ compact?

The question "Is $f(X)$ compact?" is something that occured to me when attempting the Munkres question. I think $f(X)$ is compact. Let $ \{V_\alpha \}$ be an arbitrary open cover of $Y$ such that $ ...
7
votes
3answers
153 views

Intersection of nested dense subsets.

Let $(A_n)$ be a decreasing sequence of subsets of the topological space $(X,\mathcal T)$ such that $A_1$ is dense in $X$. $A_{n+1}$ is dense in $A_n$. is $\bigcap_{n=1}^\infty A_n$ dense in $X$?
2
votes
1answer
62 views

Closed graph implies boundedness?

Let $f:\mathbb{R^n}\to \mathbb{R}$. If $G_f$ is the graph of $f$ and $G_f$ is closed, does it imply that $f$ is bounded?
3
votes
1answer
533 views

Lower limit topology regular space $ T_3$

I'm beginning to study topology and I have troubles solving these exercises (my book has no answers, unfortunately). Could you help me? Consider lower limit topology $ \mathcal{T}$ generated by basis ...
2
votes
1answer
94 views

Rotation is a homeomorphism

Fix $\alpha\in \mathbb{R}$. Let $X$ be the interval $[0,1]$ with points $0$ and $1$ identified. Define $f:X\rightarrow X$ such that $f(x)=x+\alpha\mod 1$. I need to show that $f$ is a homeomorphism. ...
3
votes
2answers
102 views

If two open sets are distinct, do they necessarily have distinct boundaries?

Is it true that for all topological spaces $X$ and all open $A,B \subseteq X$, it holds that if $A \neq B$, then $\partial A \neq \partial B$? What about if $A$ and $B$ are instead assumed closed? I ...
2
votes
2answers
369 views

Lower limit topology is a Hausdorff space $T_2$

Could you help me with the following? Consider lower limit topology $\mathcal{T} $ with basis $\mathcal{B} = \{ [a,b) \ | \ a,b \in \mathbb{R}, \ a<b \}$ Show that $\forall x,y \in \mathbb{R}, ...
2
votes
1answer
183 views

Lower limit topology closed disjoint sets

I'm currenctly studying topology and I was wondering if you could help me with the following: $X \subset \mathbb{R}$ Prove that if $A, B \subset \mathbb{R}$ are closed in $\mathcal{T}$ which is ...
3
votes
2answers
154 views

Second Countability of Euclidean Spaces

Sorry I know this is a stupid question. However I got stuck on this for quite a while. I'm trying to prove that Euclidean spaces have a countable base, which can be constructed by taking all the open ...
2
votes
1answer
108 views

showing subset of $l^2$ closed

I was looking for an example of a bounded and closed set which is not compact. Considering $l^2$ and looking to a set $K$ of canonical basis $e_i=(0,...,1_i,...,0)$. This is bounded, Is it true that ...
1
vote
1answer
110 views

Topology: functions, continuity, etc.

Let $(X,T)$ be a topological space, and let $Y$ be a set (not necessarily a topological space). Define a topology $U \in Y$ as follows: a set $U$ is defined to be open in $Y$ provided the inverse ...
27
votes
8answers
9k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
1
vote
2answers
302 views

Difference between closure and the boundary

I'm having a hard time distinguising the difference between the boundary and the closure of sets. They seem so similar, but that almost sounds too good to be true. So if the boundary is just the ...
1
vote
1answer
66 views

Please give an example in $ \mathbb{R}^{n} $ of a set that satisfies the upper bound in the Kuratowski Closure-Complement Theorem.

Please give an example in $ \mathbb{R}^{n} $ of a set $ T $ that satisfies the upper bound of $ 14 $ in the Kuratowski Closure-Complement Theorem. Thanks!
1
vote
0answers
555 views

Collection of open intervals $(a,b) \ a,b\in \mathbb{Q}$ is a basis for euclidean topology on $\mathbb{R}$

I'm not sure if this question hasn't already been asked here, but I couldn't find it. I'm currently studying topology and I'm reading a book which unfortunately has no answers to the exercises. ...
2
votes
2answers
163 views

$ \text{int}(A) = \text{int(cl}(A))$, where $A$ is convex

How can one prove that $ \text{int}(A) = \text{int(cl}(A))$, where $ A \subseteq \mathbb{R}^n$ is convex?