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
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31 views

Could someone please explain these two theorems from Topology? [on hold]

I'm trying to understand the concept of box and product topology..and these two theorems confuse me. I'd really appreciate it if someone could explain these two theorems in a simple way. Thanks!
0
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1answer
35 views

What is the interior of a single point in a metric space?

Let $(X,d)$ be a metric space. We know that if $x \in X$ , then $Cl(\{x\})=\{x\}$, which implies that $\{x\}$ is closed. However if that's the case, what would the interior of $\{x\}$ be? I was ...
2
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0answers
26 views

A compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary

Under what conditions is it true that a compact $n-1$ dimesional manifold embedded in $\mathbb{R}^n$ has no boundary (or more generally, when a manifold is embedded in some topological space)? For ...
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0answers
20 views

Marking Integers Using a Wheel

Suppose I had a wheel of diameter one meter and I was charged with marking every meter along an infinite stretch of a beach. The strategy is to insert pegs into the wheel so that every point that is a ...
1
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1answer
22 views

Prove that set of isolated points in $X$ is dense in $X$

Let $A=\{\text{isolated points of } X\}$. $X$ is a countable complete metric space. Show that $A$ is dense in $X$. My attempt: Basically we want to show that $\bar A = X$. First, we show that ...
1
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1answer
46 views

A miscellanea of properties of the rational sequence topology on $\mathbb{R}$

For each $x\in \mathbb{R}-\mathbb{Q}$ fix a sequence of rational numbers $(y_i(x))_{i\in \mathbb{N}}$ which converges to $x$. For each irrational point $x$ and each $n \geq 1$ let $M_n(x) = \{x\}\cup ...
-1
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0answers
14 views

Let $X$ be a normal space then there exists a continuous map $f : X → [0, 1]$ such that $f^{−1} (0) = A$ and $f^{−1} (1) = B$ [duplicate]

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that given $A, B ⊂ X$ closed, $∃$ a continuous map $f : X → [0, 1]$ ...
1
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2answers
23 views

Difference between Path, Curve, Graph and Trace

I am having difficulties in understanding the differences between these concepts. We have a new lecturer who loves writing down things in dense mathematical notation (I don't think that's bad but I am ...
2
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1answer
26 views

About connected topological subgroup

I'm trying to understand a proof of a theorem but I didn't understand a point. Let $G$ be an locally compact abelian group. Denote $G_0$ the connected component of $0$ (the identity of $G$). It's an ...
4
votes
1answer
45 views

Quotient topology by identifying the boundary of a circle as one point

The following is an example taken from Munkres topology book: I don't understand why does $X^{*}$is homeomorphic to $S^{2}$, is this a basic fact that I don't understand or is it an example of ...
1
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0answers
16 views

Order topology is regular and not normal

π-Base shows that linear order topology is not normal. But I remember in class the prof said order topology is normal. If $X$ is a set with linear order $<$, define a topology on X by letting ...
0
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2answers
34 views

$\{\infty\}$ open in $\mathbb N\cup\{\infty\}$ with $d(a,b)=|\arctan a-\arctan b|$?

Let $X=\mathbb N\cup\{+\infty\}$. I want to find two metrices inducing different topologies. Let $d_1$ be the discrete metric then all subsets of $X$ are open. (in particular $\{+\infty\}$) But now ...
1
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1answer
19 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
2
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0answers
12 views

Is the set of non-degenerate symmetric matrices with signature (p,q) simply connected?

Let $M=\big\{A\in\text{GL}(n,\mathbb{R})|\;A^T=A\;,\; A \text{ has signature }(p,q)\big\}\;$ denote the set of real non-degenerate symmetric matrices with signature $(p,q)$, where $p$ is the number ...
1
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2answers
49 views

Every Cauchy Sequence in the real number line converges

Prove that every cauchy sequence in $\mathbb{R}$ converges proof: Let ($a_n$) where $n\in \mathbb{N}$ be a Cauchy sequence. Let's first prove that it is bounded. Choose $\epsilon = 1$, then for some ...
0
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0answers
42 views

homeomorphism between zero-dimensional Hausdorff and two-point space

Given $\left\{g_{\alpha}: \alpha\in T\right\}$ consists of continuous functions from $A$ to $\{0,1\}$ ($A$ is a zero-dimensional Hausdorff space). Let $G =\prod_{\alpha\in T} g_{\alpha}: A\rightarrow ...
0
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1answer
23 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
2
votes
2answers
34 views

Identifying the two-hole torus with an octagon

I am aware that the 2-hole torus can be identified with the octagon with the equivalence relation as given in this picture: ...
1
vote
3answers
28 views

Show that the set of isolated points of $S$ is countable

Let $S$ be a subset of $\mathbb{R}^n$; show that the set $I$ of isolated points of $S$ is countable. Let $\mathbf{x}\in I$. There exists an open ball, say $B(\mathbf{x},r_\mathbf{x})$, of radius ...
0
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1answer
13 views

Application of Urysohn's Lemma to non-disjoint closed sets

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that: (a) Given $A \subset X$ closed, $\exists$ a continuous map ...
2
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1answer
20 views

“Closed interval” on ordered topology

Let $X$ be linearly ordered by a relation $\leq$. Taking as a subbase for topology on $X$ all sets of the form $\{x;x<a\}$ and $\{x;x>a\}$, for $a\in X$. Can be $\{x\in X; a\leq x\leq b\}$ a ...
1
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1answer
26 views

Is the saturation of Borel sets Borel?

Problem. Let $G\times X\rightarrow X$ be a continuous action of a Polish group on a Polish space. Let $A\subseteq X$ be Borel. Is the saturation $[A]_{G}:=G\cdot A$ a Borel set? One approach. The ...
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1answer
40 views

Let $X$ be a normal space then there exists a continuous map $f : X → [0, 1]$ such that $f^{−1} (0) = A$

Let $X$ be a normal space with the property that every closed set in $X$ is a countable intersection of open sets in $X$. Then show that given $A \subset X$ closed, there exists a continuous map $f : ...
0
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0answers
21 views

Precompact and locally finite implies finite intersection

An exercise in Lee's Introduction to Smooth Manifolds asks the following: Let $M$ be a topological manifold, and let $\mathcal U$ be an open cover. Suppose the sets in $\mathcal U$ are precompact ...
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1answer
19 views

Give a example of a sequence of continuous functions which do not form a Cauchy sequence

As an example that not every Cauchy sequence in $(M,d)$ is converging in $M$ the following examples are given: Consider $(\mathbb{Q},d_{\text{eucl}})$ and a sequence $q_n \in \mathbb{Q}\to ...
0
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1answer
20 views

Is Alexandroff duplicate compact?

Consider the Alexandroff duplicate $X\times_{ad} 2$, the space $X\times 2$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X$, $(U\times\{0,1\})\setminus (x,1)$ is ...
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1answer
37 views

Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology.

Find a quotient map $f:(0,1) \rightarrow [0,1]$ where the intervals $(0,1)$ and $[0,1]$ are in $\mathbb{R}$ and endowed with the subspace topology. I am really not to sure where to start. I know ...
0
votes
1answer
57 views

Finding a homeomorphism between $ B $ and $ B \times B $.

Let $ B = A^{\mathbb{N}} $, where $ A $ is a topological space. Show that $ B $ is homeomorphic to $ B \times B $. My failed attempt: I’m trying to show that there exists a function $ f: B \to B ...
9
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2answers
70 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
2
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1answer
69 views

Showing a function $f:X \rightarrow Y$ is continuous

I am working through some practice questions, and I am not sure if I am on the right track with this one: Let $X = \cup_{n≥1}A_n$, be a topological space and assume that a map f : X → Y is such ...
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
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0answers
11 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
66 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
35 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
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1answer
10 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 ...
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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
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2answers
54 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
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1answer
33 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
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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
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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
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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 ...
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0answers
16 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
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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 $d(x, y) < \epsilon$.
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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
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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
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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
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0answers
40 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
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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 ...