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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2
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1answer
14 views

Continuty of functions inside a open ball

Let $ f: X \subset \mathbb{R}^p \to \mathbb{R}^q $ and $ a \in X$. Supose that for all $ \epsilon > 0 $ exists $ g: X \to \mathbb{R}^q $ continuous at $a$ such as $ \| f(x) - g(x) \| < \epsilon ...
2
votes
3answers
33 views

if A is connected, is $\bar{A}$ connected?

where $\bar{A}$ is the closure of $A$ Here's my attempt at trying to prove this: Suppose that $\bar{A}$ is disconnected. Then, there exists open, disjoint, non empty subsets $U, V$ such that $U \cup ...
0
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2answers
22 views

is $\delta$-compact set complete?

We define $\delta$-compact metric space as monotone union of compact sets. $M=\bigcup M_i$ ($M_i\subset M_{i+1}$), is it complete?
0
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0answers
13 views

Embeddings into space equipped with final topology

Is there a nice way to check if the mappings $f_\alpha:X_\alpha\rightarrow Y$ are embeddings when $Y$ is equipped with the final topology? Or, to keep things simpler, just if they are open when ...
0
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3answers
20 views

Closure, interior and boundary of $(0, 1)$ with Zariski topology

If we consider $\mathbb{R}$ together with the Zariski topology, what is the closure, interior and boundary of $(0,1)$? A set is closed iff it is either finite or $\mathbb{R}$ under this topology, so ...
1
vote
1answer
6 views

Degrees of Polynomials that Converge Uniformly to a Non-Polynomial

I'm reading Neal Carothers' Real Analysis and there's a problem I'm stuck on: Let $p_n$ be a real-valued polynomial of degree $m_n$, and suppose that $(p_n)$ converges uniformly to a real-valued, ...
1
vote
1answer
24 views

partial converse of existence of covering spaces

suppose $X$ and $Y$ are two spaces and let $Z$ be a covering spaces for both $X$ and $Y$, then is it true that there is some space $W$ such that $X$ and $Y$ is a covering space of $W$??? (sorry but I ...
0
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2answers
20 views

How do you define the open interval $(a\times b , c\times d)$ for $a<c$; or when $a=c$, for $b<d$?

On page 85 of Topology, Second Ed., by Munkres, he draws the open interval $(a\times b , c\times d)$. I found it a little counterintuitive. Can you explain why is it that way? And why is it not ...
1
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1answer
14 views

Extending homotopies to contract an ascending union of contractible spaces

I want to extend a continuous map $H\colon X×[0..1) → X$ to a homotopy $X×[0..1] → X$ by setting $H\lvert X×[1] → X$ as the projection to $X$ (put otherwise, $H_1 = \mathrm{id}_X$). I have that $X$ ...
0
votes
1answer
18 views

Proof that the composition of two contractions on the same metric space (X,d) is also a contraction

I am required to prove that given the metric space $(X,d)$, a contraction $T : X \to X$ and another contraction $S : X \to X$, the compositions $T \circ S$ and $S \circ T$ are also contractions. I ...
0
votes
0answers
17 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
1
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2answers
14 views

Question on decreasing sequence of sets

I can't imagine this is a new question, but I was unable to find what I was looking for. I have seen it stated that if $X$ is a topological space, and $(A_{k})_{k \in \mathbb{N}}$ is a non-increasing ...
1
vote
1answer
20 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
0
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0answers
26 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
2
votes
1answer
28 views

Every isomorphism on a separable Banach space has a completely invariant dense subset

If $T$ is an isomorphism acting on a separable Banach space, can we always find a countable dense subset $D$ of $X$ such that $T(D)=D? $
1
vote
2answers
46 views

If $A,B$ are proper subsets of connected space $X,Y$, then the set $(X\times Y)\setminus (A\times B)$ is connected

Let $A$ be a proper subset of $X$ and let $B$ be a proper subset of $Y$. Prove that $(X\times Y)\setminus (A\times B)$ is connected whenever $X$ and $Y$ are connected?
1
vote
1answer
34 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
-1
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0answers
15 views

how to show the map $f(x_1,x_2) = (x_1+x_2, x_1-x_2)$ is a surjective isometry for specified metrics on $\Bbb R^2$

Question: Show that the map $f:(\Bbb R^2, d^1)\to(\Bbb R^2, d^\infty)$ defined by $$f(x_1,x_2) = (x_1+x_2, x_1-x_2)$$ is a surjective isometry. $d^1(x,y)$ is the taxi-cab metric: ...
3
votes
1answer
41 views

Finding distance between the unit ball in $\mathbb{R}^2$ and the point $(1,1)$

Given the euclidian metric. $d(x,y) = ((x_1-y_1)^2 + (x_2 -y_2)^2)^{1/2}$ find the distance between the point $(1,1)$ and the set $A = \{x=(x_1,x_2) \in \mathbb{R}^2 : x_1^2 +x_2^2 \leq 1 \}$ Where ...
6
votes
4answers
63 views

Are $\emptyset$ and $X$ closed, open or clopen?

It is indeed a very basic question but I am confused: (1) In an 2013 MSE posting under general topology here, I was told that $\emptyset$ is an open set and therefore I assume $X$ must be open too. ...
1
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2answers
19 views

Compact subset of a Hausdorff space is closed.

I'm having a bit of difficulty understanding the proof that in a Hausdorff space, a compact subset is closed. The proof I'm looking at uses the fact that a finite intersection of open sets is open ...
2
votes
1answer
23 views

Preservation of inequality on continuous functions

Let $ f,g:M \subset \mathbb{R}^{p} \to \mathbb{R} $ countinuous function at $a \in M$. Show that if $f(a) < g(a)$ then exists $ \delta >0 $ such as for $x$ and $y$ in $M \cap B(a, \delta) $ ...
0
votes
0answers
37 views

Bases for Topological Spaces Involving $I = [0,1]$

I am working on a problem where I am supposed to compare the following three topologies: The product topology on $I$$\times$$I$. The dictionary order topology on $I$$\times$$I$. The topology ...
0
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0answers
26 views

Calculate the diameter of the unit ball in $\mathbb{R}^3$ using the Euclidean metric.

So the question states, Let $B = \{x = (x_1,x_2,x_3) \in \mathbb{R}^3: x_1^2 +x_2^2 +x_3^2 \leq 1 \}$ be the unit ball in $\mathbb{R}^3$. Compute the diameter of $B$ for each of the following metrics. ...
0
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0answers
15 views

Functional Analysis/ Toplogy

The Mackey Arens theorems gives us the existence of the Mackey topology $\tau(L^{\infty},L^1)$ which is the strongest locally convex topology we can put on $L^{\infty}$ in order to make $L^1$ the dual ...
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0answers
35 views

Mathematical Expositions on Motivation

I wanted to ask this question, However I hope that it is not too soft for this site. What I want to ask is if writing an exposition on motivation for a topic of mathematics would be relevant? I.e., ...
1
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1answer
10 views

Separability of the Space of all Real-Valued functions over $[a,b]$ with a Continuous First Derivative

I'm reading Neal Carothers' Real Analysis and I'm stuck on the following question: Let $f$ be real-valued, continuously differentiable function over $[a,b]$ and let $\epsilon>0$. Show that there is ...
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0answers
36 views

Prove that $S^{n}*S^{m}=S^{n+m+1}$ [on hold]

Prove that $S^{n}*S^{m}=S^{n+m+1}$ where $*$ is the join operation on the spheres. I think that it's intuitively clear why it's true but I don't know a formal proof of this .
0
votes
1answer
22 views

Is there a countable basis for the finite complement topology on the natural numbers?

I have the following question on a worksheet: Consider the finite complement topology τf on N. Does this topology have a countable base? If so give one such base and if not prove your claim. Been ...
0
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0answers
8 views

An intuitive affirmation about convex sets - normal at the boundary of a convex set

Let $\Omega_1 \subset \Omega_2$ two open bounded sets in $R^n $with $\Omega_i$, $i=1,2$ convex and with $\overline{\Omega_1} \subset \Omega_2 $. Suppose that $\partial \Omega_2$ is $C^1$. Now fix $y ...
1
vote
1answer
29 views

A question on Lebesgue's covering dimension

Roughly, a compact, Hausdorff space $X$ has covering dimension $\leqslant n$ if each finite cover $\mathcal{U}$ of $X$ can be refined by a cover $\mathcal{V}$ such that each point $x\in K$ belongs to ...
2
votes
0answers
36 views

Both $F$ and $C$ are closed sets but their sum $F+C$ is not closed. [duplicate]

In context to the question what will be an counter example such that both $F$ and $C$ are closed sets in $ \Bbb R^n$ but their sum $F+C$ is not closed in $ \Bbb R^n$?
0
votes
2answers
24 views

First axiom of countability and finer topologies

If $(X,T)$ and $(X,T')$ are topological spaces, where $T\subseteq T'$, then if $(X,T)$ satisfies the first axiom of countability, not necessarily does $(X,T')$ and viceversa. However, I am not able to ...
1
vote
1answer
50 views

A problem about homotopy equivalence in Hatcher's book

In reading the proof of corollary 0.21 of Hatcher's algebraic toplology. I can not understand how the existence of homotopy equivalence $f:X\rightarrow Y$ implies that the inclusion $X\rightarrow ...
0
votes
1answer
23 views

A space homotopy equivalent to its subspace implies the inclusion map is a homotopy equivalence?

I find it not easy to understand the proof of corollary 0.21 of Hatcher's algebraic toplology. If the question of the title is true, I can understand it. But I don't know how to prove it. Can somebody ...
1
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0answers
25 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
1
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0answers
15 views

homotopy between continuous functions to an absolute retract

I have the following statement to prove as one of the "fundamental" questions our topology professor wants us to know for his final: Let $X$ be a topological space, and let $A$ be an absolute ...
1
vote
2answers
51 views

What does it mean to attach a cell to a space by a map?

I am starting to study for my algebraic toplogy exam and a lot of the problems sound like: let $Y_p$ be the space obtained by attaching an $(n+1)$-cell to $S_n$ by a map of degree $p$, where $p$ ...
1
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1answer
38 views

Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s. So here's what I know so ...
3
votes
1answer
41 views

Example of colimit of Hausdorff spaces which is not Hausdorff

In http://mathoverflow.net/questions/195248/co-hausdorffification, it is mentioned that the subcategory of Top consisting of Hausdorff spaces is not closed under colimits. The simplest colimit I ...
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0answers
24 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
0
votes
1answer
16 views

Base for the Topology Generated by a Family of Semi-norms?

Let $\mathscr{P}$ be a family of semi-norms on a $\mathbb K=\mathbb R$ or $\mathbb C$ vector space. Can anyone help me showing the collection $$\mathscr{B}:=\left\{\bigcap_{j=1}^n B_{p_j}(x, ...
1
vote
1answer
27 views

Wheel Graphs and Dimension of Embeddings

I'd like to preface this by saying this is the tip of the iceberg for an optional question for a summer REU program application, so if you think asking this question is in bad taste, let me know and I ...
2
votes
2answers
31 views

Specific Question About Open/Close Sets.

So I had a question about open/close subsets because we started this in Topology today. So let's take an closed subset of $\mathbb{R}$ so, for example $X = [-2, 2]$. Let's say I want to compare if a ...
1
vote
1answer
39 views

Different definitions of subnet

I encountered two different definitions of subnet. The first is Let $(I, \preceq_I ), (J,\preceq_J )$ be two directed sets and $X$ be the underlying set.$\{ \eta_j \}_{j \in J}$ is a subnet of $\{ ...
0
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1answer
31 views

Deforming $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$

I am trying to find a deformation retraction of $\text{id}: S^1 \to S^1$ to the symmetry $S^1 \to S^1$ such that $x \mapsto -x$. I guess this deformation of maps has to respect all homotopy rules, ...
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0answers
10 views

Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
-1
votes
0answers
31 views

Mackey Topology

Let $C$ be a convex subset of the unit ball of $L^{\infty}$. Show that if $C$ is closed in the topology induced by the standard $\|\cdot\|_p$ norm for some $p>1$, then $C$ is closed in the Mackey ...
1
vote
1answer
22 views

Complete metric space, fixed point and ?reverse? fixed point theorem.

Let $(X,d)$ be a complete metric space, let $F: X\rightarrow X$ such that $$\exists L > 1, \forall (x,y)\in X^2, d(F(x),F(y))>L\cdot d(x,y).$$ Show that if $F(X)=X$ then there is exactly one ...
-1
votes
1answer
34 views

Questions of Hyperspace of Compact Sets

Let $K(X)$ the space of all non-empty compact subsets of $X$ equipped with the topology from the Hausdorff metric. if $X$ is metrizable and $K_n\in K(X)$, $K_1 \supseteq K_2 \supseteq \ldots$. Then ...