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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1answer
41 views

How to solve following problem in topology involving relatively compact sets?

For a subset $A$ of a topological space $X$, let $\hat{A}$ denote the union of set $A$ and all those connected components of $X\setminus A$ which are relatively compact in $X$. Then for every $A \...
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0answers
12 views

The range of a continuous function on the order topology is convex

Let $(W, \leq)$ be a linear order, and let $f : [0, 1] \rightarrow W$ be a continuous function (where [0, 1] has the usual topology and W has its order topology). Show that the range of f is convex. ...
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0answers
16 views

Does distance characterize open sets in $\Re ^n$

I was wondering if given the standard topology for $\Re ^n$ that I might be able use the concept of distance in open sets to prove certain properties of the topological space. For example if I have ...
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1answer
34 views

Filter, which does not have the Baire property

I do not understand the following proof of the following theorem. Let $\mathcal{F}$ be a filter on $\omega$ which contains the cofinite filter. (1) For every partition of $\omega$ into finite sets $...
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1answer
16 views

Is the unit square with dictionary ordering second countable?

I'm conflicted: If we consider the set $\{x\} \times (0,1)$, for $ x \in [0,1]$, these are open in the unit square, uncountable and disjoint, but what about open intervals of the form ((a,b), (c,d)) ...
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0answers
19 views

Extending real valued to analytic functions

Is there any way to extend a real valued function to analytics functions on a open set? I would think not since if $f$ and $g$ are real valued functions their difference $f - g$ is also a real valued ...
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2answers
40 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open.

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
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1answer
12 views

Properly discontinuous group actions - Hausdorfness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
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0answers
36 views

Hatcher's exercise 1.2.22 on the Wirtinger presentation

Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations $...
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1answer
30 views

Let $X$ be a metric space and $Y$ be a open set where $Y=\cup_{x \in Y} B(x, r(x)) $ The union in this theorem will have to be infinite. Why?

Let $X$ be a metric space and $Y$ be a open set($Y \subseteq X$). Now, according to a theorem, $Y=\cup_{x \in Y} B(x, r(x)) $ i.e. $Y$ as a union of open balls. The union in this theorem will have to ...
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29 views

Topology of CW-complex and attaching map

I think I must have a fundamental misconception in place right now in my mind. When defining a CW-complex, we use inductively continuous maps from $f_{\partial \sigma} :S^n \to K^{(n)}$. We then ...
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24 views

Finding the boundary of the continuous image of a compact simply-connected Lie group

Statement of the problem Given a continuous map $f:G \rightarrow D^2$ where $G$ is a compact simply connected Lie group and $D^2$ is the unit disk in the plane, I have shown that: There exists a ...
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88 views
+50

Can $ f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $f^{-1}(y) = \{a_y,b_y\} $ be continuous?

This is the problem we want to solve: Can $f\colon \mathbb{R}^k \to \mathbb{R}^n$ such that $ \forall y \in \operatorname{im}(f)$, $ f^{-1}(y) = \{a_y,b_y\}, a_y \neq b_y $ be continuous? ...
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1answer
28 views

Show that the set of the radii of the disks in $S$ is bounded

Let $(b,c)\subseteq\Bbb R$ be an open interval. $S$ is a set of disjoint closed disks in $\Bbb R^2$, such that $\forall s\in S \ \exists x\in (b,c)$ such that $(x,0)\in s$. Show that the set of the ...
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1answer
25 views

Baire property and $F_\sigma$ sets

I am currently reading a proof for a theorem which concludes that a set A does not have the Baire property, if (1) is correct. So, $(1)\Rightarrow$A does not have the Baire property is shown. The ...
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1answer
49 views

Proving $(f \circ g)_{*}= f_{*} \circ g_{*}$

I have two continuous functions $f: X \to Y$ and $g: Y \to X$ such that $f \circ g =id_Y$ and $g \circ f =id_X$. If I induce maps $f_{*}$ and $g_{*}$ everywhere I search tells me that it's clear that $...
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0answers
13 views

How many connected components can the intersection of a ball with an arbitrary domain have?

Let $\Omega \subseteq \mathbb{R}^n$ be an arbitrary domain (i.e. a connected and open subset). Let $x \in \Omega$ and $B_r(x)$ be the ball (w.r.t. euclidean metric) around $x$ with radius $r$. How ...
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0answers
15 views

Best open approximations of the diagonal

Suppose $A$ is a topological space and $\Delta_A\subseteq |A|\times |A|$ is the diagonal relation. $\Delta_A$ is rarely open (indeed it is open if an only if the topology on $A$ is discrete), but it ...
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1answer
41 views

I want to prove that $B=S^2 \cap \{(x_1, x_2, x_3) : x_i \geq 0 \}$ is homeomorphic to the disk $B^2= \{ x \in \mathbb{R^2} : ||x|| \leq 1 \}$

I claim that the map $f:B \to B^2$ defined by $f(x_1, x_2, x_3)=(x_1, x_2, 0)$ is a continious, bijection with $g: B^2 \to B$, $g(x_1, x_2)=(x_1, x_2, \sqrt{1-(x_1^2+x_2^2)})$ it's inverse. So i ...
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0answers
25 views

What are “boundaries” (as defined here) really called and where can I learn more?

My guess is that boundaries (perhaps under a different name) in graph theory are probably defined like this: Definition 0. Let $G$ denote a graph, $A$ denote a subset of $G$. Then a candidate ...
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3answers
19 views

Show that $f$ is continuous in the relative topology

Let $(X, T)$ be the subspace of $\mathbb R$ given by $X = [0,1]\cup [2,4]$. Define $f: (X,T) \to \mathbb R$ by $$f(x)= \begin{cases}1 & x \in [0,1] \\ 2 & x \in [2,4]\end{cases}$$ Show that $f$...
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1answer
51 views

Graph is closed $\iff$ $f$ is Continuous

Let $X$ be a metric space and $Y$ be a compact metric space. I want to show that the graph of $f$, $G\subset X\times Y$ is closed $\iff$ $f:X\to Y$ is continuous. I'm not sure where the compactness ...
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0answers
18 views

Pictures and illustrations of attaching a 2-cell to a wedge of circles?

I am having a lot of trouble visualizing how to attach a disk "along a word". Where can I find some pictures and illustrations of this procedure?
5
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1answer
1k views

Graph of continuous function from compact space is compact.

I know this question seems to have been asked hundreds of times, but I don't really see how any of the existing answers address my concern, so I'm hoping that maybe someone here might be able to ...
9
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1answer
1k views

$f$ continuous iff $\operatorname{graph}(f)$ is compact

The Problem: Let $(E,\tau_E)$ be a compact space and $(F,\tau_F)$ be a Hausdorff space. Show that a function $f:E\rightarrow F$ is continuous if and only if its graph is compact. My Work: First ...
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2answers
25 views

Every sequence in a compact set $E$ has at least one accumulation point

Proof by contradiction. Say there is no accumulation point, therefore $$\bigcap_{k \in \mathbb{N}}\Big(\overline{\bigcup_{n \geq k}\{x_n\} }\Big) = \emptyset$$ This is equivalent to $$E = E \ \...
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1answer
24 views

In topological space $(X, \mathcal{O})$ with $\emptyset\neq A\subseteq X$ do we have $X\setminus\partial A=X^\circ$?

I got that result by basic set operations and that $X \setminus \bar{A} = (X \setminus A)^\circ$ but am not sure if it is correct. $ X \setminus \partial A\\ = X \setminus (\bar{A} \setminus A^\circ)\...
5
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1answer
109 views
+50

Milnor's definition of smooth manifold

In Milnor's book "Topology from a differential viewpoint" on page one he defines a smooth manifold to be a subset $M \subset \mathbb R^n$ which is locally diffeomorphic to some open subset of $\mathbb ...
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1answer
144 views
+100

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
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4answers
61 views

Doubt in a step in a proof

Let $G$ be an open set in $\mathbb{C}$. Let $\gamma:[a,b]\to G$ a rectifiable path. In a step in a proof it is stated that since $\{\gamma\}$ is a compact set (the braces denote the image set of $\...
4
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2answers
48 views

Is $\bigcup_{x \in A} [x - 1, x + 1]$ Lebesgue measurable, where $A$ is a Lebesgue measurable subset of $\mathbb{R}$?

Suppose $A$ is a Lebesgue measurable subset of $\mathbb{R}$ and $$B = \bigcup_{x \in A} [x - 1, x + 1].$$Is $B$ Lebesgue measurable?
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0answers
29 views

Existence of separating hyperplane after injective mapping of compact set

Let $f: X \to Y$ be an injective continuous function where $X \subset \mathbb{R}^m$ is a nonempty compact set, and $Y \subset \mathbb{R}^n$. Let $\mathbf{y}^* = f\left(\mathbf{x}^*\right)$. If $E \...
3
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2answers
47 views

Preimage of open set is Lebesgue measurable only if the function itself is measurable

It is a simple result in my book saying the proof is trivial, but I can not seem to show it. If someone can provide a hint just to help me begin my proof, it would be of assistance. Assume you know ...
0
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0answers
29 views

Let m = $min(\omega_1)$. Show that {m} is clopen in $\omega_1$

Let $(W, \leq)$ be an uncountable well order, then $\omega_1 = \{\alpha \in W: pred(\alpha)$ is countable } where $pred(\alpha) = \{y \in W: y < \alpha \}$ I have a couple of thoughts and I was ...
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2answers
35 views

Problem in understanding the proof of closure of topological closure is closure.

Well, I was following the proof of Closure of Topological Closure equals Closure; here is the proof: It follows directly from Set is Subset of its Topological Closure that: $$\overline{H} \...
3
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4answers
56 views

Where is the flaw in my reasoning in this proof that $\overline{A \cap B} = \overline{A} \cap \overline{B}$?

Let $X$ be a topological space, and $A,B, \subset X$, then since the closure of a set may be characterized as as the smallest closed set containing the set, certainly we have $A \subset \overline{A}$ ...
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1answer
22 views

verify the meaning of the closure of a set in a finite/non-metric space.

My motivation behind this question is to better under the fundamental concepts of topology in their general sense. I understand what a set closure when dealing with real numbers and a euclidean ...
2
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1answer
14 views

Ascending chain of monotone classes, $A$ necessarily in $\mathcal{M}$

Suppose $\mathcal{M}_1 \subset \mathcal{M}_2 \subset \ldots$ are monotone classes. Let $\mathcal{M} = \bigcup_{n = 1}^\infty \mathcal{M}_n$. Suppose $A_j \uparrow A$ and each $A_j \in \mathcal{M}$. Is ...
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0answers
31 views

Is $\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}$ a $\sigma$-algebra of subsets of $X$ or not?

Let $(Y, \mathcal{A})$ be a measurable space and let $f$ map $X$ into $Y$, but do not assume that $f$ is one-to-one. Define$$\mathcal{B} = \{f^{-1}(A) : A \in \mathcal{A}\}.$$Is $\mathcal{B}$ a $\...
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1answer
69 views

Meager subset of $2^\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
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0answers
47 views

Looking for a right book for Algebraic Topology - is Dieck's textbook a good choice?

I self-study Algebraic Topology. I use Hatcher's textbook Algebraic Topology and soon I'm going to end reading Chapter 3. I know that there is one more chapter about homotopy theory but I'd like to ...
4
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1answer
68 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
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0answers
59 views

Is there a continuous path through the complement of a family of closed sets in the unit square, if none touches both bottom and top?

I wonder whether the following claim is correct. Consider a family of connected closed subsets of the unit square that (i) do not touch each other and (ii) contains its closure, in the sense that ...
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2answers
18 views

Example of a set and two $\sigma$ algebras such that union is not a $\sigma$-algebra

What is an example of a set $X$ and two $\sigma$-algebras $\mathcal{A}_1$ and $\mathcal{A}_2$, each consisting of subsets of $X$, such that $\mathcal{A}_1 \cup \mathcal{A}_2$ is not a $\sigma$-algebra?...
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0answers
21 views

Example of a set and monotone class where monotone class is not a $\sigma$-algebra?

What is an example of a set $X$ and a monotone class $\mathcal{M}$ consisting of subsets of $X$ such that $\emptyset \in \mathcal{M}$, $X \in \mathcal{M}$, but $\mathcal{M}$ is not a $\sigma$-algebra?
4
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1answer
41 views

Special chains in $\mathbb{R}^n$

We had a topology exam yesterday, where the following was a question: Prove or disprove that there is a chain (ordered by set inclusion) of discrete subsets of $\mathbb{R}^n$ containing uncountably ...
3
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1answer
89 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
5
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1answer
92 views

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
0
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1answer
27 views

Dimension of $C^n$, space of all $n$-differentiable functions?

What is the proper classification for the (infinite) dimensionality of $C^n$, the space of all functions (defined on $\mathbb{R}^m$, $m\in\mathbb{N}$) with continuous derivatives from order 0 to $n$? ...
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0answers
17 views

Reference Ascoli-Arzelà Theorem

I am looking for a reference in the literature of the following corollary of Ascoli-Arzelà's theorem : $K\subset\mathbb{R}^n$ is compact. The set $\{f:\mathbb{R_+}\to K \ | \ f \text{ is }1- \text{ ...