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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Question about quotient of a compact Hausdorff space

I am reading the book 'Algebraic Topology' by Tammo Tom Dieck. On page 12 in the proposition 1.4.4 he states that : Let $X$ be a compact Hausdorff space and $f : X \rightarrow Y$ be a ...
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83 views

Adherent values for a sequence in a metric space

I have these two definitions for an adherent value of a sequence the first is : $a$ is a an adherent value for $(x_n)$ iff $$\displaystyle \forall \varepsilon>0,\forall n\in \mathbb{N},\exists ...
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1answer
69 views

The set $A := \{ (x,y) : y \in f(x,x+1) \}$ is closed if $f : \mathbb{R} \to \mathbb{R}$ is continuous.

Let $f : \mathbb{R}\to \mathbb{R}$ continuous. I need to prove that $A := \{ (x,y) : y \in f(x,x+1) \}$ is closed in $\mathbb{R^2}$. We have that $\mathbb{A}$ is the union of all the ...
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1answer
151 views

A problem in A Course in Point Set Topology by Conway, union of totally bounded sets

This is stated as a problem in A Course in Point Set Topology book by J. Conway: Let $\{E_n\}$ be a sequence of totally bounded sets. If $\operatorname{diam}E_n\to 0$ as $n\to\infty$, show that $\...
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1answer
68 views

null homotopy of 1-skeleton of simply connected simplicial complex induced by homotopy identity

I am reading a book suggesting that the following is true: Let $X$ be a simply connected (maybe finite) simplicial complex. Then there exists a continous map $$g : X \to X,$$ homotopic to the ...
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1answer
69 views

Show that the set is a closure of $U_j$

Let $\Omega$ be an open and nonempty set and $\Omega \subset \mathbb{R}^n$. Let's define a set $U_j=\{x\in\Omega:\Vert x\Vert<j \wedge dist(x,bd(\Omega))<\frac{1}{j}\}$. We observe that $U_j$ ...
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1answer
165 views

Jingle River (Berbed wire) Metric Problem

I want to prove that Jungle River metric is indeed a metric space, and determine it is open and closed balls. Firstly, i know that the metric is given by $x,y\in \mathbb{R}^2$, such that $x=(x_1,x_2), ...
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2answers
298 views

Closed or open + proof

Proof if $H = \{ (x,y)\mid 0<x<1, 0<y<1\}$ is closed, open or neither. So I would strongly suggest that it is open because if you draw it you recognize that the complementary set ...
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2answers
55 views

A question about two common definitions

Two definitions make me puzzled ! 1. The definition of $\textbf{Functions Differentiable at a Point}$: A function $f$ defined in a neighborhood $(x_{0}-\delta,x_{0}+\delta)$of a point $x_{0}$, ...
3
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1answer
244 views

Definition of continuity in topological spaces does not seem quite right.

Here's how continuity is defined in most standard topology texts A function from $X$ to $Y$ is continuous iff the inverse image of each open set of $Y$ is open in $X$. This definition does not ...
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1answer
49 views

Topologial properties of $f([1,3]^3)$ where $f(x,y,z)=x^2+2xz+y$

If $f(x,y,z)=x^2+2xz+y$, determine $f([1,3]^3)$ and characterize this set in terms of openness, closedness, completeness, compactness and connectedness. Since $[1,3]^3$ is compact then $f([1,3]^3)$ ...
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1answer
41 views

Decompostion into countably many nowhere dense compact sets

Let $A$ be a meagre subset of a locally compact abelian Polish group $G$. Then $A$ can be written as a countable union of nowhere dense subsets of $G$. Is it always possible to write $A$ as a ...
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2answers
48 views

Opens in a manifold

I am studying for my exam of manifolds and I dno't understand why the following is true. Let $(U,\phi)$ be a local map at $p$ in the manifold $M$, and let $\phi(p)=0$. Then there exists an open set $...
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2answers
796 views

Proof that there is no continuous 1-1 map from the unit circle in $\Bbb R^2$ to $\Bbb R$.

Let $S^1=\{(x,y) \in \Bbb R^2 : x^2+y^2=1\}$. I'm trying to prove that there is no 1-1 continuous mapping between $S^1$ and the real line. The map is not necessarily onto. Proof so far: Suppose such ...
3
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2answers
62 views

Open/Closed value of unions of Open/Close intervals.

I hope I do not make a duplicate here but I couldn't find this question on here nor a clear answer on the internet. So let's say I have two intervals, $A$ and $B$. Let's define $C$ as the union of $A$...
2
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1answer
108 views

Definition of disconnected subsets in metric spaces and in more general settings

I found the following paragraph in a Real Analysis book (namely, Carother's one). A subset $E$ of a metric space $M$ is disconnected in $E$ if there exist disjoint, nonempty, open (in $E$) ...
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2answers
68 views

could a spanning tree graph be expressed by a lower triangular matrix?

Suppose a directed spanning tree graph $G$, there are $n$ nodes, and the root is node $1$. We express this graph by a matrix $M_{n\times n}$. If there is an directed edge from node $i$ to node $j$, ...
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1answer
67 views

Which is finer(larger) between the sequence spaces $l_{p}$ & $l_{p+1}$

Prove that, $l_{3}\subset l_{7}$ & $L_{9}[0,1]\subset L_{6}[0,1]$, where $l_{p}$ & $L_{p}[0,1]$ are of usual notation. Are the converses hold for both cases? Can these two results ...
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1answer
73 views

Level set of a continuous function strictly increasing in each argument

Let $F : \mathbb{R}^d \to [0,1]$ be absolutely continuous and strictly increasing in each argument. Is it true that the boundary of the set $\{ \boldsymbol{x} \in \mathbb{R}^d: F(x) \geq \alpha \}$ ...
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1answer
58 views

Show there is a path in X $\times$ Y if and only if there is a path in X and a path in Y

(a) Let $(x_1,x_2) \in X$ and $(y_1,y_2) \in Y$. Show that there is a path from $(x_1,y_1)$ to $(x_2, y_2)$ if and only if there is a path $x_1$ to $x_2$ in $X$ and a path $y_1$ to $y_2$ in $Y$. (b) ...
0
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1answer
48 views

the openness of the union of two disconnected sets (one of which is open and the other one is closed)

Now we have two disconnected sets in complex plane, one of which is open and the other one is closed. Is the union of these two sets open?
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1answer
64 views

Does $B(H)$ satisfy in Heine-Borel property?

Based on here, I know that every bounded and closed subset of a space is not compact. I really want to know that $B(H)$, the space of bounded linear operators, satisfies in Heine - Borel property. ...
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1answer
106 views

Is it possible for a manifold to have a normal vector that is zero everywhere, if so, would this indicate that the manifold is non-orientable?

Basically I've been thinking about defining a non-orientable three-dimensional metric space via defining the normal vector and looking to see if there is two possible vectors for the same point. I'm ...
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0answers
31 views

Upper hemicontinuity of a correspondence

I would like to know whether the following correspondence is upper hemicontinuous: $$ C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases}, $$ ...
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1answer
61 views

Why $X$ contains a countable $\pi$-basis?

I don't understand the following statement. First, I write what a $\pi$-bases means. Let $X$ be a topological space and $\mathcal{B}$ a family of non-empty open sets. We call $\mathcal{B}$ a $\pi$-...
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0answers
69 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
3
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1answer
185 views

Motivating the compact-open topology

It has been a while since I studied algebraic topology, and I wanted to revisit homotopy theory. Determined to take a more sustainable approach, I started by questioning and verifying every result in ...
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2answers
202 views

What is the one point compactification of the reals?

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a ...
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1answer
243 views

Limit points and boundary sets in topology

The main difference between an open set and a closed set is a closed set includes its boundary while an open set does not. However, in topology, a closed set is also distinguished (from an open set) ...
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2answers
158 views

Closure of an open set in manifold

I have a question. During a proof of a proposition, the following is stated: Let $K$ be a compact set in a manifold $M$ of dimension $n$. Then there exists an open set $U$ such that $K\subset U$ and ...
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1answer
11 views

balls have empty boundary with regard to the $p$-adic norm

Let $p$ be prime, $a\in\mathbb{Q}$ and $r\geq0$. How can I show that the closed ball $D(a,r)$ in $(\mathbb{Q},|\cdot|_p)$ must have an empty boundary (with regard to the topology induced by the $p$-...
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0answers
50 views

Space generated by a reflection

Suppose I embed a mirror (not necessarily plane) in some space (say a manifold). Is there a theory that tells you how to classify the "space" generated by the reflection (the one you see if you were ...
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1answer
36 views

does closed under sequence limits imply the set is closed?

In a topological space $T$ we have a set $F$ such that the limit of every convergent sequence of elements of $F$ is in $F$. can we deduce that $F$ is closed? if $T$ be second countable then it's true,...
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2answers
50 views

Topology on k((t))

$k((t)):=\lbrace (a_i)_{i \in \mathbb{Z}}, a_i \in k,\exists \ N \in \mathbb{Z} \ s.t \ \forall \ i<N, a_i=0\rbrace$ where $k$ is a field of char zero. We define componentwise addition and ...
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484 views

Let X and Y be topological spaces. A function f: X → Y is continuous if and only if $f^{-1}$ (C) is closed in X for every closed set C ⊂ Y.

I need help proving this theorem This is the first part of the biconditional, I think if I can prove this. Proving the converse shouldn't be nearly as difficult. Assume f is continuous ⇒ $f^{-1}$ (C)...
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2answers
72 views

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$. When is $f(K)=[f(a),f(b)]$? What I believe is the ...
2
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1answer
74 views

Prove that Open Sets in $\mathbb{R}$ are The Disjoint Union of Open Intervals Without the Axioms of Choice

There are several proofs I have seen of this, but they all seem to use choice subtely at some point. Is there any way to prove this without choice, or is it possibly unproveable?
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2answers
67 views

Manifold with $\pi_1(M)=F_n$

We may construct a 3-manifold $M_n$ with $\pi_1(M_n)\cong F_n$ (i.e. the free group on $n$ generators) as follows: consider the complement of $n$ pairs of open 3-balls in $\mathbb{R}^3$. For each pair,...
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1answer
62 views

When does a topological group embed topologically in its group of homeomorphisms?

Let $X$ be a topological group. $X$ acts freely on itself by left multiplication; this gives us an injective group homomorphism $\Phi: X\rightarrow \operatorname{Homeo} X$. Under what conditions is $\...
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1answer
61 views

Every point of an open ball is a centre for the open ball.

Suppose $X$ is a nonempty set and $d$ is an ultrametric on $X$ i.e.,$$d(x,y)\le\max\{d(x,z),d(z,y)\}$$ for all $x,y \in X$. Suppose B is an open ball of $(X,d)$. Show that every point of B is a ...
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0answers
59 views

How to prove a function is harmonic polynomial

1! How to prove this function a harmonic polynomial using Laplace equation For the 1 question I know we can prove harmonic using Laplace Equation but for this on m confused how to start. For the 2 ...
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2answers
754 views

Is sum and product of a infinite number of continuous functions are also continuous functions?

Whether in Real Analysis or by Open Set Def of Continuity in Topology, it is easy to show that the sum and product of a FINITE number of continuous functions are also continuous functions. That is, ...
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51 views

Is the question in the Munkres's topology book wrong?

At the end of cheapter $8.1$, $4)$ Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. $b)$ Show that if $Y$ is path connected, the set $[I,Y]$ has a ...
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1answer
67 views

Dense set meaning on this proof…

Theorem: Let $f$ and $g : X → Y$ be continuous functions (Open Set Definition of Continuity). Assume that $Y$ is Hausdorff and that there exists a dense subset $D$ of $X$ such that $f(x) = g(x)$ for ...
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3answers
36 views

$S^{n+m+1}$ can be decomposed as the union of $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ along their boundaries.

Let $S^n$ denote the $n$-sphere and $D^n$ denote the $n$-disk (of course, $\partial D^{n+1}\cong S^n$). Then $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ both have boundary $S^n\times S^m$. The ...
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1answer
212 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
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2answers
193 views

Continuous function on complete bounded metric space need not be bounded

I came across the following old qual problem: Suppose $(X,d)$ is a complete metric space with finite diameter. Is every continuous function on $X$ bounded? It seems like the function $1/x$ on ...
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1answer
38 views

Find the map of the closed ball $B(0,1)$ of the following continuous function $f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1)$ and $f^{-1}(0)$.

Find the map of the closed ball $B(0,1)$ of the following continuous function $$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right)$$ and $f^{-1}(0)$. $f^{-1}$ seems quite simple, I got $(0,2,-9)$,...
0
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1answer
322 views

Show that the intersection of two connected sets is connected if the two sets are disjoint.

Show that the intersection of two connected sets is connected if the two sets are disjoint. Is the set $1\leq x^2+y^2+z^2 \leq 9$ connected and/or compact? I think its compact because it's closed ...
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1answer
120 views

Modern treatment of Topology that focuses on intuition and is full of explanations and visual insights.

I'm interested in a modern treatment of Topology (point-set, and general topology at the undergraduate level) that focuses on intuition and is full of explanations and visual insights. This will be ...