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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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., ...
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1answer
7 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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33 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 .
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19 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 ...
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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 ...
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1answer
26 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 ...
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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$?
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1answer
15 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 ...
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26 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 ...
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21 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 ...
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17 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 ...
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12 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 ...
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2answers
44 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 $p$ ...
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1answer
27 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 ...
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1answer
37 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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21 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 ...
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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, ...
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1answer
25 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 ...
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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 ...
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75 views

Question about computing a Complicated integral

where $\beta$ is defined like this: I'm trying to prove (2.18) but i don't know how to do, i calculated the integral but i don't find anything %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EDIT1: ...
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1answer
34 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 $\{ ...
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29 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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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 ...
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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 ...
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1answer
21 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 ...
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33 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 ...
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1answer
45 views

Do $X\times Y$ and $X_c\times Y$ have the same compact subsets?

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
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1answer
25 views

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ into the constant map $\mathbb{R}^2 \to \mathbb{R}^2?$

Can we deform continuously $\text{id}:\mathbb{R}^2 \to \mathbb{R}^2$ where $(x,y) \mapsto (x,y) $, into the constant map $\mathbb{R}^2 \to \mathbb{R}^2$ (where I guess $(x,y)\mapsto (0,0)\,\,$)?$ I ...
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1answer
37 views

Attaching maps in a product cell complex

This is a follow-up to those two questions: Cartesian product of two CW-complexes, and Product of CW complexes question. Consider two cell complexes $A$ (with cells $e^m_\alpha$ and attaching maps ...
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17 views

Proper holomorphic maps and the degree of the map

Suppose f is holomorphic and maps U onto V, both being disks. If f is proper, does this induce a well defined degree for f? And does the converse hold? What are some tools that can help me see if ...
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15 views

Two non-homeomorphic spaces with continuos bijective functions in both directions

I was asked the following question: if two topological spaces $X, Y$ are such that there exist a function $f:X\rightarrow Y$ continuos and bijective and a function $g:Y\rightarrow X$ continuous and ...
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63 views

A question about CW complex

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
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35 views

Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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32 views

A question about compactly generated topology [on hold]

Given a space $X$ and a collection of subspaces $X_\alpha$ whose union is $X$, these subspaces generate a possibly finer topology on $X$ by defining a set $A\subset X$ to be open iff $A\cap X_\alpha$ ...
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1answer
28 views

Order topology on a subset may be weaker (but never stronger) than the subspace topology

If $X$ is a linearly ordered set, the topology $\mathcal{T}$ generated by the sets $\{x:x<a\}$ and $\{x:x>a\}$ ($a \in X$) is called the order topology. Suppose $Y$ is a subset of $X$, show ...
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1answer
20 views

Converse of the Uryshon metrization theorem

Uryshon metrization theorem says that every regular and second countable topological space is metrizable. My question, is the converse of this theorem ture ? If not, what are the counter examples? ...
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1answer
12 views

from each one-third part that eliminated in construting the Cantor set pick a point, what apout the resulting set?

During constructing the cantor set, pick up a point from the one-third that eliminated. if we call the set of this points A, then what is the internal of A? is the complement of A countable?
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1answer
31 views

Characterisation of the order topology on a linearly ordered set

If $X$ is a linearly ordered set, the topology $\mathcal{T}$ generated by the sets $\{x:x<a\}$ and $\{x:x>a\}$ ($a \in X$) is called the order topology. If $a, b \in X$ and $a<b$, there ...
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27 views

Is a countable product of open intervals homeomorphic to $\mathbb{R}^\omega$?

Fix countably many intervals $(a_i,b_i) \subset \mathbb{R}$, and let $\pi_{i \in \mathbb{N}} (a_i,b_i)$ be their Cartesian product with the product topology. Question: is $\pi_{i \in \mathbb{N}} ...
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1answer
30 views

how to prove matrix addition is continuous under certain matric topology?

let $A,B$ be $m \times n$ matrices . $\|A\|$ := the square root of sum of (individual entry square) (hope it's clear :P) $d(A, B) = \|A − B\|$, already proved that $d$ is a metric. (1)now proved ...
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Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
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2answers
33 views

Every $p$-norm ($p \in [0,\infty]$) generates the same class of open sets on $\mathbb{R}^n$

The following claim has been made in my multivariable analysis class, and I think I have the idea of the proof but I can't quite seem to get down to the rigorous proof the instructor wants: Every ...
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1answer
33 views

Derivatives in Topological Vector Spaces and General Spaces

I know that we may have a function $f$ such that all the directional derivative of $f$ in some topological vector space $V$, but the derivative need not exist. Moreover, it's possible for all the ...
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1answer
82 views

Why is Klein bottle non-orientable?

I am doing the homework of differential geometry and encounter this problem: The Klein bottle $K^2$ is defined to be the identification space $$[0, 1] \times [0, 1]/{\sim}, \text{ where the ...
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1answer
21 views

If $X$ is a set and $\tau_1$ is finer than $\tau_2$, prove if $(X, \tau_2)$ is Hausdorff, then $(X, \tau_1)$ is Hausdorff.

I tried to do this by contradiction. So we have that $(X, \tau_2)$ is Hausdorff, and $\tau_2 \subset \tau_1$. Suppose that $(X, \tau_1)$ was not Hausdorff. Then we have elements $y,z \in X$ where ...
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1answer
34 views

Is this strengthening of paracompactness known?

Consider a topological space $X$. What can be said about the following property? For any open cover $\mathcal U = \{ U_i \}_{ i \in I }$ of $X$, there exists an open refinement $\mathcal V = \{ V_j ...
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26 views

Topology over $C^0(\mathbb{R})$

Let $C^0(\mathbb{R})$ be the set of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$, For any continuous function $h > 0$ consider $B_f(h) = \{ g \in C^0(\mathbb{R}) : |f(x) - g(x) ...
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41 views

To show that $X = (0,1]$ is complete .

Show that $X = (0,1]$ is complete with respect to the metric $e $ where $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. My proof: let $(x_n)$ be Cauchy in $(X,e)$. Let $(t_n) := \frac{1}{(x_n)}$. Then ...
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25 views

Homology of Subspace vs. Homology of Ambient Space.

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle ...
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1answer
28 views

Non-homeomorphic (?) subspaces of Euclidean plane

Let $Y_1 = \bigcup_{n=1}^{\infty}I((0,0),(\frac{1}{n},\frac{1}{n^2}))$ and $Y_2 = \bigcup_{n=1}^{\infty}I((0,0),(1,\frac{1}{n})) \cup I((0,0),(1,0))$ where $I$ denotes line segment in Euclidean space. ...