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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Find all regions formed by a set of circles

I was doodling with Python to draw some circles, and I was able to find all intersection points of a set of random circles, yay ! Now I'm stuck on a question, is there a way to find all regions ...
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29 views

The plane minus a countable set homeomorphic to the plane minus an uncountable set?

Is it possible that $\Bbb R^2-C$ can be homeomorphic to $\Bbb R^2-U$ where $C$ is countably infinite and $U$ is uncountable? Intuitively I believe the answer is no, but I'm having difficulty ...
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62 views

A few questions on the properties of $\mathbb{R} ^ {[0,1]}$

Given the topological space $X=\mathbb{R}^{[0,1]}$ with the product topology, there are several properties regarding to $X$ which I am not sure if are true/false. Is $X$ metrizable? I'm having ...
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18 views

Show that $H= \cup _{r \in \mathbb {Q } \cap [0,1 ] }(K+r) $ is bounded, where $K $ is compact .

I want to show that $H= \cup _{r \in \mathbb {Q } \cap [0,1 ] }(K+r) $ is bounded, where $K + r $ denotes the translate and $K $ is compact . This is sort of obvious but I want to construct an ...
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107 views

is the lexicographic order topology on the unit square connected/path connected?

I was wondering, given the lexicographic order topology on $S=[0,1] \times [0,1]$, is it connected (and path connected)? I found a reference to Steen's and Seebach's Counterexamples in Topology, and ...
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23 views

Determining the connectedness of $\{(x,y,\sin(x^2+y^2)) : x^2+y^2=1\}$

This question is on my exam review sheet. It says we can use the fact that $\{(x,y) : x^2+y^2 = 1\}$ is connected. Am I correct in saying $f : \mathbb{R}^2 \to \mathbb{R}^3$, $f(x,y) = (x,y,\sin(1))$ ...
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67 views

Is every linear ordered set normal in its order topology?

I'm trying to prove (or disprove) that every linear ordered set $(X, <_X)$ is normal in its order topology. I was able to prove $(X,<_X)$ is hausdorff, simply by taking two open intervals with ...
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29 views

Topology, small detail on a proof. Concept Closure and Adherent

I'm trying to recreate the proof that: If $Y $ is a subset of a metric space $ X $, then the closure of $ Y $ is closed. But I cannot provide a proof of the following: Let $ x \in \overline{\overline ...
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2answers
70 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
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32 views

The plane minus the graph of a continuous function consists of two path-connected components?

Let $f:\Bbb R\rightarrow \Bbb R$ be continuous. Show that $\Bbb R^2-\mathrm{graph}(f)$ consists of two path-connected components. I can show that the area 'above' the graph of $f$ and the area ...
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34 views

Showing that the image of a polynomial map is not closed

Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers. Then for some irreducible ...
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46 views

The nonemptiness of the intersection of compact sets such that all finite intersections are nonempty

From Rudin's Principles of Mathematical Analysis: Theorem 2.36: If {$K_\alpha$} is a collection of compact sets of a metric space X such that the intersection of every finite subcollection of ...
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1answer
40 views

Complement of closed dense set

Let $X$ be a topological space and $C$ be its closed and dense subset. Then is it possible for $X-C$ to be dense in X? I think $C$ doesn't have to be closed, and in that case $X-C$ can be also dense. ...
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13 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
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20 views

Question about Hausdorff spaces and their equivalences [duplicate]

Definition: A topological space $X$ is called Hausdorff space if for each $x_1,x_2 \in X$ (they are distinct) we can always find neighborhoods $U_1,U_2$ of $x_1,x_2$ such that $U_1 \cap U_2 = ...
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52 views

Show that the set is compact using the definition

The set in question is $\{0\}\cup \{1,\frac12,\frac13,\ldots,\frac1n,\ldots\}$ (for $n\in\mathbb N$). Okay, so for a set to be compact, every open cover of it must be able to be broken down into a ...
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50 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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1answer
24 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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128 views

When is the free loop space simply connected?

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I'm wrong, but ...
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27 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
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1answer
48 views

MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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49 views

Open/closeness of subsets of natural numbers

So I've just started reading about neighbourhood and Hausdorff space. It makes me wonder if $(\mathbb{N},\mathcal{P}(\mathbb{N}))$ is Hausdorff and why, and are sets in $\mathbb{N}$ open or closed or? ...
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36 views

Limit points Topology

I'm trying to prove the following: Th: A subset of a topological space is closed iff it contains all of its limit points. Defn of a limit point of a subset $A$ is the following: $p \in X$ is a limit ...
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36 views

equivalent characterisation of simply connect spaces

I want to prove the following: Let $X$ be path connected space, $S^{1}$ the $1$-sphere and $D^{2}$ the unit circle. Following are equivalent: i)X is simply connected. ii) If $f:S^{1} \to X$ ...
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85 views

The Fundamental Group - An explicit homotopy between $(f \circ g) \circ h$ and $f \circ (g \circ h)$

I'm wondering if anyone can help me to understand a proof that the fundamental group is in fact a group. I am looking at the proof on page 3 of this document. I understand everything, although I am ...
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30 views

Lattice Version of Stone-Weierstrass

I've been reviewing Stone-Weierstrass theoerem. While reading the wikipedia page I read the following version of the theorem: Suppose $X$ is a compact Hausdorff space with at least two points and $L$ ...
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38 views

Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of ...
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1answer
59 views

Tietze Extension Theorem

I saw Tietze extension theorem. Since its proof is non-trivial, I tried whether we can clarify it intuitively for functions of one real variable. So, in this special case, I am trying to prove that if ...
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157 views

Is the plane minus the integer lattice homeomorphic to the plane minus the integers?

The question, more rigorously posed, is: Is $\Bbb R^2-\Bbb Z^2$ homeomorphic to $\Bbb R^2-\Bbb Z\times\{0\}$? This question has been bugging me in the back of my head for a while now. Sometimes, ...
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41 views

determine if set is open or closed

I have to determine whether the sset {1,2,3} is open or closed. I have never done these types of questions before but this is what I did (on pic). please can I have some feedback if I have done it ...
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1answer
71 views

Invariance of domain in $\mathbb{R}^2$

Let $U \subseteq \mathbb{R}^2$ an open subset and let $f:U\rightarrow \mathbb {R}^2$ is be a continuous function. I have the following version of Invariance of Domain Theorem (in $\mathbb{R}^2$): If ...
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16 views

Compact frames, an equivalent reformulation

$\top$ denotes the greatest element of a poset. Adapted from nLab: Definition 1. A frame is compact is and only if for every collection of opens whose union is $\top$ (which covers $\top$), there is ...
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46 views

A question regarding a discrete sub-space

denote by $A$ the set $\{\frac{1}{n} \,\, |\,\,\, n\in N \}$. Is $A$ a discrete sub-space of $\mathbb{R}$? (with the standard topology) I was told that it is, but I think it isn't. For example, ...
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22 views

finite simplicial complex compact

Let $K=(V,\Sigma)$ be a finite simplicial complex. I want to show that $|K|$ is compact. I know that $K$ is a sub-simplicial complex of $\Delta^V$ with $|\Delta^V|$ compact. So I think I should show ...
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33 views

What is the maximal size of an equal-distance set in $\mathbb{R}^n$?

Let $A\subseteq \mathbb{R}^n$ with the casual metric and $c\in\mathbb{R}^+$ be a real positive number, such that for every $a_1, a_2\in A$ if $a_1\neq a_2$ then $d(a_1,a_2)=c$. What is the maximal ...
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56 views

Is $\omega_1$ metrizable?

Following Urysohn's metrization theorem, I would like to prove or disprove that $\omega_1$ is metrizable. I know it is hausdorff, but I'm not sure whether or not it is second countable, and I'm at ...
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93 views

An open connected subset of compact metric connected and locally connected space, is path connected

I need to prove: If $X$ is compact, metric, connected and locally connected space, and $U$ is open connected subset of $X$, then $U$ is path connected. Using the following: a) If $X$ connected, ...
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31 views

The space of $S^1/S^1$, the space of a single point, and their first homotopy group

I read from the book Soft matter physics by Kleman that the space $R$ of a point is $0$ and its first homotopy group $\pi_1(0)=0$. This causes some confusion to my understanding. Why the space of a ...
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1answer
37 views

Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
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81 views

Involution on Cantor space with exactly one fixed point

Let $X=\{0,1\}^{\mathbb{N}}$ be the Cantor space. What is an example of a continuous map $\sigma : X \to X$ with $\sigma^2=\mathrm{id}$ and $\# \{x \in X : \sigma(x)=x\} = 1$? This has to exist, ...
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67 views

Proving that $ω_1$ is locally compact

I'm trying to show that $ω_1$ is locally compact, but when doing so, I need to show something else, which got me a bit stuck on. I'm taking a $\alpha\in ω_1$, so $\{\alpha\}$ is an open set. Since ...
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129 views

Covering $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments

Is it possible to cover $\mathbb{R}^2$ with uncountably many disjoint non-degenerate line segments? If a formal definition is necessary, let's define a line segment as a set $\{(x, mx+c): x \in [a, ...
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1answer
23 views

Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
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37 views

Is there a quotient map between arbitrary topological spaces?

Let $X,Y$ be topological spaces. If there is a quotient map $p:X\rightarrow Y$, then the topology on $Y$ is completely determined by $p$. I'm curious whether the converse holds. That is, if we know ...
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48 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...
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1answer
39 views

Break up $\mathbb{R}P^2$ into a part homeomorphic to Mobius band & part homeo. to the 2-disc

The claim is that $\mathbb{R}P^2 = A \cup B$ where $A \simeq$ Mobius band, $B \simeq D^2$, and $A \cap B \simeq S^1$. I understand this intuitively with a gluing type argument, similar to the ...
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16 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
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67 views

Tangent bundles of smooth manifolds

Using the identity $T(M \times N) = T(M) \times T(N)$, it is easy to construct the tangent bundles for various smooth manifolds such as the n-dimensional sphere $S^{n}$. However, I could not figure ...
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57 views

If $P$ denotes the Cantor set, then show that $[0, 1] \setminus P$ is dense in $[0, 1]$

I have done the following. Suggest if the Proof is rigorous enough. Let us select any point $x\in [0,1]$. Ternary expansion of $x$ can be represented by $x=0.b_1 b_2 b_3 b_4\ldots$ where ...
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22 views

A question about the dimension of topological products

For each positive integer n, is the (small inductive) dimension of the topological product of n copies of the "long line", always equal to n? I ask because the "long line" is not a separable metric ...