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
56 views

Countable dense subsets of $\mathbb R$ are homeomorphic

Suppose countable subsets $A,B$ of the real line $\mathbb R$ satisfy $\overline{A}=\overline{B}=\Bbb R$. How can one show that $A$ is homeomorphic to $B$? I even have no idea how to get a bijection ...
3
votes
0answers
12 views

Existence of a closed and open set in 0-dimensional Hausdorff space [duplicate]

Given $T = 0-$dimensional (i.e, T has a basis of sets which are both open and closed), Lindelof Hausdorff space, with closed subsets $A$ and $B$ such that $A\cap B = \emptyset$. Prove that $\exists$ a ...
1
vote
2answers
67 views

Metric spaces are completely normal

Given a metric space $(X, k)$ with $Y, Z\subset X$ and $\operatorname{cl}(Y)\cap Z = \emptyset$, $\operatorname{cl}(Z)\cap Y = \emptyset$, prove that there are open sets $M, N$ such that $Y\subset ...
1
vote
1answer
36 views

Fiber bundle beginner question.

I'm reading some notes on fiber bundles. Let $f:X \rightarrow Y$ be a continuous map of topological spaces. The author states: We say $f$ makes $Y$ a fiber space over $X$ if $f$ is locally trivial ...
0
votes
1answer
11 views

Constructing an almost contained set from a family of sets with strong finite intersection property.

I don't even know if this is true but I have a feeling I've read it's true somewhere. A counterexample or a proof would be equally welcome, or a link to where I can find more information. (Maybe the ...
1
vote
0answers
21 views

$c_{00}$ is a dense subset of $c_0$

I would like to show that $c_{00}$ is a dense subset of $c_0$. I am not sure if I am overly simplifying the argument or even making the right argument for that matter. proof: Suppose that $x \in ...
2
votes
2answers
59 views

In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Prove that if $X$ is Hausdorff and $\mathfrak{C}$ is a nonempty chain of compact and connected subsets of $X$, then $\bigcap \mathfrak{C}$ is compact and connected. Here are the definitions which ...
1
vote
1answer
34 views

Prove that a non-empty subset of an open set which is evenly covered is evenly covered

Let $p: E\rightarrow B$ a continuous surjective map and $U \subseteq B$ be open and not empty and who is being evenly covered by $p$. Show that all non-empty subsets of $U$ are being evenly covered by ...
0
votes
0answers
37 views

When is a product of [0,1] separable? [on hold]

I need help proving the following: $[0,1]^A $ is separable iff |A|$\leq 2^{\aleph_0}$
0
votes
1answer
23 views

Homeomorphism and Split Interval

Let us consider the split interval $S(I)$: that's the space $I\times 2$ endowed with the topology generated by the lexicographic order. We can consider, analogously, the space $S(2^\omega)$ and delete ...
0
votes
1answer
16 views

Is $\left(\bigcap_{i=1}^{\infty}A_i\right)^{o} = \bigcap_{i=1}^{\infty}A_i^{o}$?

$A^{o}$ is the set of all interior points. The definition of an interior point is as follows: Let $A$ be a set of real numbers. A point $p\in A$ is an interior point if and only if $p$ belongs to some ...
0
votes
1answer
27 views

Is $\left(\bigcup_{i=1}^{\infty}A_i\right)^{o} = \bigcup_{i=1}^{\infty}A_i^{o}$?

$A^{o}$ is the set of all interior points. The definition of an interior point is as follows: Let $A$ be a set of real numbers. A point $p\in A$ is an interior point if and only if $p$ belongs to some ...
1
vote
0answers
17 views

Characterizing equicontinuity via ultrafilters

We have a compact metric space $(X,d)$ and a homeomorphism $T:X\to X$. For any ultrafilter $p\in\beta\mathbb{Z}$ we can define the map $T^p:X\to X$ given by $T^p(x):=\lim_{n\to p}T^n x$ (which can ...
0
votes
1answer
24 views

How do you prove that a metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property below?

A metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property: for $\epsilon > 0$ and every $x \in X$, there is a $y \in Y$ such that $d(x, y) < \epsilon$.
5
votes
2answers
130 views

Understanding the definition of nowhere dense sets in Abbott's Understanding Analysis

First of all, I am sorry for asking a question about understanding a definition in a book named Understanding Analyis. But it is my first time to encounter basic topology, so I hope you can excuse me. ...
0
votes
0answers
43 views

Give an example for which $\lim X_n$ does not exist but $\limsup X_n$ does?

Can anyone give me an example of a sequence $X_n$ for which $\lim X_n$ does not exist but $\limsup X_n$ does? Thanks!
2
votes
0answers
27 views

Spaces in which the closure of every countable subset does not include an uncountable closed discrete subset

What classes of spaces $X$ have the property that that for every countable subset $C \subset X$, $\overline{C}$ does not have an uncountable closed discrete subset? I know every space with countable ...
2
votes
0answers
41 views

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed.

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed. I have found an example for the map not to be closed. But unable to prove that it is open. Please ...
0
votes
1answer
21 views

What are the neighborhoods of f(x)

In general, does a neighborhood of a function g(x) have to contain only elements $y$ such that there exists an $x$ for which $f(x)=y$? More concretely... I'm trying to learn topology from Munkres' ...
0
votes
2answers
22 views

Negate a proposition with quantifier?

I'm going over the proof of the theorem stating that "In a metric space, compactness impliess sequential compactness". I'm very likely confusing myself. I have the following proposition: $\forall ...
-4
votes
1answer
52 views

Problem of Topology by Dugundji, chapter III, section 4 [on hold]

Prove: G is open in X if and only if $\overline{G \cap \overline{A}}$ = $\overline{G \cap A}$ for all $A \subseteq X$.
1
vote
1answer
29 views

Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
3
votes
0answers
36 views

Initial and final topologies

Suppose that $X_i$ are topological spaces, and $X_i \xrightarrow{f_i} Y$ are a family of maps into the set $Y$. The final topology on $Y$ is defined to be the finest topology on $Y$ such that each ...
0
votes
2answers
48 views

A metric space on which every real-valued function is continuous

Let $(X,d)$ is a metric space such that every arbitrary function $f:X\to\Bbb{R}$ is continuous. Then which option is right? a) $X$ is bounded. b) Every subset of $X$ is closed. c) Every subset of $X$ ...
0
votes
1answer
25 views

Defining an action on $\mathbb{R}\times[0,1]$ such that the orbit space is homeoomorphic to the Möbius band?

By Möbius band I mean the quotient space obtained from $[0,1]^2$ by identifying (0,y) with (1,1-y). Not syre how to show that a given orbit space is homeomorphic tl a given space.
3
votes
0answers
51 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
2
votes
2answers
21 views

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open.

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open. To show the map $f : X \to Y$ is open, we first take any open subset $U$ from $X$ and then show that ...
2
votes
0answers
31 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
0
votes
0answers
19 views

Good partition in locally compact metric space

Let $X$ be a locally compact second countable metric space, endowed with a Borel probability space $\mu$. Let $\varepsilon>0$. Question: Is it possible to find a countable Borel partition $\xi$ of ...
0
votes
1answer
22 views

Homeomorphism of a triangle.

Let $\triangle$ be any equilateral triangle with its interior embedded in $\mathbb{R}^2$. Given that $f:\triangle\rightarrow\triangle$ is an homeomorphism. How can I prove that $f$ is an isometry? I ...
4
votes
1answer
60 views

Do join and suspension commute?

Do join and suspension of topological spaces always commute, i.e. is it true that $\sum(A\star B)=A\star(\sum B)$? I suppose that it is not true in general (but, for example, everything works in the ...
1
vote
1answer
24 views

Is $\overline{\mathbb{R}}^+$ a compact Polish space

if $X$ is defined by $$X= [0,+\infty)\cup\{+\infty\}$$ is endowed with the metric $$d_X(x,y) = |\arctan(x) - \arctan(y)|$$ Is it true that the metric space $(X,d_X)$ meets the following properties? ...
0
votes
1answer
25 views

Did I just prove $h^{-1}(W)$ is open in $X$?

Let $h:X \rightarrow Y$ be a function between topological spaces. Let $U$ be a closed subset of $X$ and $g=h|_{U}$ be the restriction. Suppose further that $g$ is continuous. Let $W$ be open in $Y$. ...
0
votes
1answer
15 views

Show that $\Pi X_n=X$ is totally bounded under $D$ (Munkres , page 280,Q.1)

Suppose that each $X_n$ is metrizable with $d_n$, $D(\vec{x},\vec{y})=\sup\{\bar{d_i}(x_i,y_i) \}$ is a metric for the product space $X=\Pi X_n$. Show that $\Pi X_n=X$ is totally bounded under $D$. ...
2
votes
2answers
437 views

Are all smooth functions bounded?

in my book it says that when a function f is smooth, it also means that it is bounded. I understand that a smooth function has contineous derivatives of all orders, but how can we know that the ...
1
vote
2answers
31 views

Definition of continuity implies a discontinuous function is continuous?

So I have a text that defines a function $f$ to be continuous if $f^{-1}(A)$ is open whenever $A$ is open. However, that seems like a confusing definition since it doesn't specify if the open sets ...
1
vote
0answers
37 views

$(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$

Let $X_1$, $X_2$, and $X_3$ be spaces. (a) Prove that $(X_1 \times X_2) \times X_3$ is homeomorphic to $(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$ So, I think I ...
1
vote
1answer
24 views

set of arbitrary positive measure conformally equivalent to unit disk

Show that for any  $\epsilon$ > 0, there is a dense subset of $\mathbb{C}$ with measure less than $\epsilon$ and which is conformally equivalent to the unit disc. To make a dense set that has ...
2
votes
0answers
47 views
+50

Cardinality of a minimal open cover of the disc

Consider $D_1^2(0)=\{x\in\Bbb R^n: ||x||_2\leq 1\}$ and let $\epsilon>0$. Consider the open cover $\mathcal{O}=\{B_\epsilon^2(x):x\in D_1^2(0)\}$ of $D_1^2(0)$. What is the minimum cardinality ...
1
vote
1answer
16 views

Checking my understanding of the Interior of these intervals

Let $[a,b]$ be any finite closed interval. (i) $\text{Int}_{[a,b]}(a,b]$ Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in ...
-1
votes
1answer
44 views

classification theorem in a subset of R^2

I need some very simple results of algebraic topology but I am not sure where I can find them without having to swallow the whole theory. What I want: -An open bounded subset $A$ of $R^2$ is ...
2
votes
2answers
31 views

closed sets which are strictly contained in open set

I wanted to ask the following: Suppose I am in the complex plane with the usual topology. Suppose I have a closed set A such that A ⊂ B, where B is an open set. I wanted to know if I can always find a ...
1
vote
1answer
24 views

Question on definition of group acting on a topological space.

I know that a group can act on a graph by acting on the set of vertices on a graph. I also know that a graph can be viewed as a CW complex and therefore a topological space and i am trying to bridge ...
1
vote
1answer
38 views

Proving that these curves intersect

Let $\Gamma$, $\Sigma$ be two curves with ranges in $(\{0\}\cup\mathbb{R}_{+})^2$. $\Gamma$ starts on the $y$ and ends on the $x$ axis: $\Gamma(0)=(0,\gamma_2),\Gamma(1)=(\gamma_1,0)$. $\Sigma$ is a ...
2
votes
1answer
29 views

Homemorphism from projective plane to S1 and Moebius strip

Let $h$ be a homemorphism from $S^{1}$ to the border of the Möbius strip $M$. Also, let $X$ be the quotient of the disjoint union of the closed unit disk $D^{2}$ and $M$ by the equivalence relation ...
0
votes
1answer
56 views

First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
2
votes
1answer
50 views

Universal covering of the complement of a circle in $\mathbb{R}^3$

What is the universal covering of $X=\mathbb{R}^3\setminus(S^1\times\{0\})$? I've been trying to build a covering map from $\mathbb{R}^3$ onto $X$ via composition of $p:\mathbb{R}^3\to Y$ and $q:Y\to ...
1
vote
1answer
28 views

An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
0
votes
1answer
17 views

$\{K∈K(X):K⊆U\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$

Let $X$ be a Polish space. The family of set $(i)$ $\{K∈K(X):K⊆U\}$ $(ii)$ $\{K∈K(X):K∩U≠∅\}$ for $U$ open in $X$ generates $\textbf{B}(K(X))$ where $K(X)$ is the space of all compact subsets of ...
0
votes
1answer
16 views

Is a complement of the union of all open first category subsets a Baire's space?

Let $X$ be a topological space and $P$ be the union of all open subsets of $X$ which are of the first category. By the Banach category theorem $P$ is of the first category. Is a closed subspace ...