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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3
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2answers
85 views

Prove that there are open set $U$ and $V$ in $X$ such that $x∈U, y∈V$ and $U∩V=∅$

Let $X,D$ be a metric space. Suppose that $x$ and $y$ are two distinct points of $X$. Prove that there are open set $U$ and $V$ in $X$ such that $x∈U, y∈V$ and $U∩V=∅$ My professor gave me a hint ...
1
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1answer
101 views

A subbase for the weak topology of real functions

I'm reading a book chapter on weak topology, where it says that For a family $\mathscr{F}$ of real functions on $X$, the weak topology generated by $\mathscr{F}$ is denoted ...
2
votes
1answer
108 views

How can I prove that this space is not Urysohns frechet space?

Define $f:\Bbb R\to Z,$ where $Z=(\Bbb R\setminus \Bbb N)\cup\{a\}$ for some $a\notin\Bbb R,$ by $$f(x)=\begin{cases}x & \text{if }x\in \Bbb R\setminus \Bbb N\\ a & \text{if }x \in\Bbb ...
0
votes
3answers
58 views

Boundedness of functions over a certain interval

I was looking at my real analysis notebook from my undergrad days, and I stumbled upon the theorem that seems a bit counter intuitive to me. I was wondering if someone could explain or perhaps offer a ...
0
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1answer
52 views

Relation between open and closed sets in Metric space

let $(\mathbb{N},d)$, where $d : \mathbb{N} \times \ \mathbb{N} \to \mathbb{R}$, $d(m,n)=| \frac{1}{n}-\frac{1}{m}| $. Is every open set closed at the same time? Find $Bd(\{n \in \mathbb{N} : n \ge ...
0
votes
1answer
118 views

General Topology - separation axiom

Recall that $a$ is an accumulation point of a set $A$ in a space X if and only if each neighborhood of $a$ meets $A$ in some point other than $a$. We say $a$ is a condensation point of $A$ if and only ...
3
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2answers
498 views

K topology: Examples

Why would the interval $(-3,1)$ be open in the $k$-topology? (I'm using Munkres). Can I have some other examples of intervals in $k$-topology? What exactly does $(a,b)$ $\cup$ $(a,b)-k$ for ...
2
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1answer
494 views

Why is the weak* topology not in general metrizable?

A Banach space is a topological group under addition. The dual is a topological group under the weak$^*$ topology. The weak$^*$ topology is weaker than the operator norm topology, so is it ...
11
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1answer
207 views

Is there an infinite connected topological space such that every space obtained by removing one point from it is totally disconnected?

The particular point topology on any set is connected, but on removing the particular point, the complement is discrete, and hence totally disconnected. Although this is not even $T^1$, Cantor's leaky ...
0
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2answers
159 views

distance between two disjoint compact subsets in a metric space

Let $X$ and $Y$ be nonempty disjoint compact subsets in a metric space $(Z,d)$. Define $d(X,Y):=\inf \{d(x,y): x\in X, y\in Y\}$. How can I show that there exist $x_0\in X$ and $y_0\in Y$ such that ...
2
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2answers
46 views

closure of the set is $\mathbb{R}^\mathbb{N}$

I have encountered this problem in the book and got stuck for quite a while now. Please help me! Let $X$ be the set of all $x\in\mathbb{R}^\mathbb{N}$ such that for some $N\in\mathbb{N}$, $x(n)=0$ ...
0
votes
2answers
88 views

Point Set Topology: Countably Infinite

I am trying to work a problem and I am having trouble wrapping my mind about what $\textbf{X-U being countably infinite mean}$?. Here X is our given set and U subset of X such that X-U is countble or ...
-1
votes
1answer
36 views

$x$ is either an element of $S$ or a limit point of $S$ if and only if every open set containing $x$ intersects $S$?

Let $(X,\mathcal{T})$ be a topological space, $S\subset X$ and $x\in X$. Can I show that: $x$ is either an element of $S$ or a limit point of $S$ if and only if every open set containing $x$ ...
0
votes
1answer
108 views

Show that every finite-dimensional topological vector subspace is closed.

Let $X$ be a normed topological vector space. Show the following: (i) If $0\neq v \in X$, then $\{\alpha v:\alpha\in \mathbb{R}\}$ is closed. (ii) If $Y$ is a closed vector subspace of $X$ and $w\in ...
1
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1answer
49 views

How can be a class of paths be open for a connected open set?

I have the following excercise: Let $A$ be an open set. If $x,y\in A$ we write $x\sim y $ when there is a path from $x$ to $y$, this is, $\exists P=\bigcup_{i=0}^{n} [r_{i-1},r_i]$ with $a=r_0$ and ...
1
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3answers
108 views

Is there a norm on ${\Bbb R}^{\Bbb N}$

Let $E={\Bbb R}^{\Bbb N}$ be the real vector space of real sequences. 1) Is there a norm on $E$? 2) Is there a norm $N$ on $E$ such that the restriction of $N$ to $\ell^2$ is finer than the ...
1
vote
3answers
186 views

Do continuous mappings always have an inverse?

A theorem of general topology states that: A mapping $f$ from $X$ to $Y$ is continuous if and only if the inverse image of any open set in $Y$ is open in $X$. Does this mean that continuous ...
2
votes
1answer
79 views

What conditions on a totally ordered set imply that the closed intervals are connected?

It is well known that $[a,b]_\mathbb{R}$ is connected, while $[a,b]_\mathbb{Q}$ is not. I'm wondering, what conditions on a totally ordered set $T$ imply that for all $a,b \in T$, it holds that ...
3
votes
1answer
116 views

Topology of the Sorgenfrey line

Does the Sorgenfrey line have the homotopy type of a CW-complex? I know that the Sorgenfrey line is a paracompact, Hausdorff space, but cannot be a manifold because this space is not locally compact. ...
2
votes
1answer
114 views

Show no open ball in $C[0,1]$ is contained in the space of all Lipschitz-continuous functions on $[0,1]$

This is a question I encountered in a textbook exercise. Here $C[0,1]$ is the space of all real-valued continuous functions on $[0,1]$ endowed with the sup metric, ...
2
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3answers
128 views

General Topology: Neighborhood

Show that a subset $U$ of the real numbers is open in the usual topology if and only if, for all $x$ in $U$, there is a number $\epsilon>0$ such that $|y-x|<\epsilon$ implies $y$ is in $U$. "I ...
1
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1answer
60 views

A topology in a Vector space

Let $V$ a finite dimensional vector space and let $T$ the collection of all finite unions of vector subspaces of $V$, with the empty set. Show that the elements of $T$ are the closed sets of a ...
3
votes
1answer
102 views

Number of topologies on a set: An expression?

Given a set $\emptyset \neq X \in\{\mathbb{N}, \mathbb{R} \} \cup \{A: |A| \in \mathbb{N} \}$, is there an expression for the number of topological spaces, $(X,\_)$, as a function of $|X|$ ? I am ...
12
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2answers
432 views

Motivation for introducing algebraic topology?

What kind of topological questions does algebraic topology answer where point set topology is not enough? Phrased differently: Where is the line (or maybe intersection) between point set topology ...
1
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3answers
128 views

Definition of a topological space

This may be a simple question, but I am nonetheless confused. I am just starting to learn about topology. I am confused by part of the Wolfram Mathworld article on Topological Spaces. In it, it is ...
0
votes
0answers
37 views

A function $f: H \to \mathbb{R}$ is not weakly continuous at $0$ but $(f(x_n)$ converges to $0$ whenever $(x_n) \to 0$ weakly in $H$

Let $H$ be a Hilbert space equipped with its weak topology and let $K \subset H$ such that $K = \{ \sqrt{n}e_n | n \in \mathbb{N_0} \}$ Let $f:H \to \mathbb{R}$ be a function such that $f(x) = 1$ when ...
5
votes
1answer
176 views

Is every $G_\delta$ set the set of continuity points of some function $f$?

I can prove that given a function $f:X \rightarrow Y$, where $X,Y$ are metric spaces, the set $A \subseteq X$ of points on which $f$ is continuous, is $G_{\delta}$. (Take $U_n = \bigcup_{y \in ...
2
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0answers
63 views
1
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0answers
46 views

Sets that have the property of Baire

Can I say that a set $A$ has the property of Baire, if and only if it is of the form $A=(B \setminus C) \cup D$ where $B$ is regular open and $C,D$ are of first Category? Are there any other useful ...
1
vote
1answer
113 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
1
vote
1answer
107 views

Why only one unbounded connected component

Here on page 344 it is stated that If $U \subset \mathbb C$ is bounded then $\mathbb C \setminus U$ has exactly one unbounded component. While it seems sort of clear to me in an intuitive way I ...
0
votes
1answer
48 views

A question on Cauchy sequence in topological abelian group

Let $G$ be a topological abelian group. Recall that a Cauchy sequence $(x_n)$ in $G$ is defined to be a sequence such that for any neighborhood $U$ of $0$, there exists an integer $N$ with ...
6
votes
1answer
244 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
10
votes
1answer
219 views

In the Sorgenfrey plane, is an open disc homeomorphic to an open square?

In the sorgenfrey plane $\mathbb{R}_l^2$, the subspace $$X=\{(x,y):x^2+y^2\leq 1\}$$ is not homeomorphic to the subspace $$Y=\{(x,y):|x|\leq 1,|y|\leq 1\},$$ because there is only one isolated point ...
2
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0answers
63 views

Examples of extremally disconnected spaces

I am trying to understand the notion of extremally disconnected space (in other words Stonean space), i.e. a space in which any open set has an open closure. Could you help me and give (reasonable) ...
3
votes
2answers
56 views

Discrete family of compact subsets in a metric space

A family $\{A_i: i \in A_i\}$ of subsets of a topological space $X$ is said to be discrete if each $x \in X$ has a neighborhood meeting at the most one $A_i$. Suppose $(X,d)$ is a metric space and ...
0
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0answers
36 views

What are the compact sets in chains or posets with either the left or interval topology?

In the real line in the usual topology the compact sets are the closed and bounded sets. In the left topology, the topology generated by the left-open intervals $(-,a)$, the compact sets are exactly ...
1
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1answer
82 views

General topology: Showing a set is open.

I am using Munkres and the problem states, let $\textbf{X}$ be a topological space: let $\textbf{A}$ be a subset of $\textbf{X}$. Suppose that for each $x \in \textbf{A}$ there is n open set ...
4
votes
2answers
198 views

Good source for a point set topological introduction to CW complexes?

Most algebraic topology books I found don't dwell too much on point set topology of CW complexes. I'd like too become more familiar with them. Anyone knows a good source (with exercises) too learn ...
1
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4answers
75 views

Prove that any subset of any countable set $S$ is countable

Prove that any subset of any countable set S is countable Here is what I got Proof: We assume that $W$ is a subset of a countable set $S$. We will show that $W$ is also countable. Since $W$ is a ...
2
votes
1answer
95 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
4
votes
5answers
148 views

Nets and Convergence: Why directed indices?

Please do read carefully (I know Nets-Topology-Filters and their interrelations!!!) 1.) Why do we require nets to be indexed by directed sets (apart from it simply works compared to filters and ...
4
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0answers
56 views

(Dis)continuity of stepfunction in topology

I'm trying to learn a little about topology, and I don't quite understand continuity yet. I use this definition of a continuous map f: f is continuous if the inverse image of every open set is open. ...
1
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2answers
178 views

Regions determined by a closed curve

Question: is "Idea" below flawed? Let $\gamma$ be a closed curve in the complex plane. It may intersect itself, and required only to be continuous (no differentiability assumptions). The image of ...
2
votes
1answer
314 views

Relationship between different topologies of bounded operators on a Hilbert space

I am self-studying functional analysis. Given that $B(H)$ are the bounded operators on a Hilbert space, $H$. I would like to ask how to formally prove that the weak topology is weaker than the ...
4
votes
1answer
78 views

Baire Category Theorem proof in Gamelin Greene - how do they shrink the closure of open ball

I am confused by a step in the Gamelin and Greene proof of the Baire Category Theorem. Here is the start of the proof. Theorem: Let $\{U_n\}_{n=1}^{\infty}$ be a sequence of dense open subsets of a ...
3
votes
1answer
99 views

Embedding 2nd countable, zero-dimensional Hausdorff space in the Cantor space

The Cantor Space $2^{\mathbb N}$ is the space of all infinite $0$-$1$-sequences with the metric $d(x,y) = 0$ for $x=y$ or $d(x,y) = 1/k$ where $k$ is the least integer such that $x_k \ne y_k$. Now I ...
2
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1answer
54 views

Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
1
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1answer
39 views

Let A be a set in the Euclidean space. Is the set A relative open with respect to its closure?

Let $A$ be a connected set in the Euclidean space $\mathbb{R}^n$. Is the set $A$ relative open with respect to its closure $\bar{A}$?
0
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1answer
71 views

is the closure of a subalgebra also a subalgebra?

Let X be a topological space and A a subalgebra of C(X,R) or C(X,C). Is the closure of A a subalgebra? Here C(X,R) and C(X,C) denote the set of all real continuous and bounded functions and the set of ...