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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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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1answer
26 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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9 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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22 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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1answer
28 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
43 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
23 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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25 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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1answer
14 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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14 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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0answers
50 views

A question about CW complex [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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31 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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30 views

A question about compactly generated topology

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
27 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
11 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
30 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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25 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
67 views

Relationship between $d(A,B)\gt 0$ and $A \cap B = \varnothing$

a) Show there exists closed non-compact subsets in $\mathbb{R}^2$ such that $d(A,B) = 0$ and $A \cap B = \varnothing$ b) Given $K$ being compact and $B$ closed, show there is a sequence $x_n \in K$ ...
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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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22 views

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
32 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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0answers
22 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
31 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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3answers
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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0answers
23 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
26 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. ...
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1answer
40 views

Prove that $\mathbb{Z}$ is a closed subset of $\mathbb{R}$ [duplicate]

Let $\mathbb{Z}$ and $\mathbb{Q}$ represent the integers and the rationals, respectively. (a) Prove that $\mathbb{Z}$ is a closed subset of $\mathbb{R}$. ...
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1answer
19 views

describe/sktech a picture of b1((0,0)).

I have attached an image of the problem I need help with b and c B i know a circle with a ball centered at the orgin and the set of points less than 1. and c how do i use triangle inequality?
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1answer
26 views

$T$ is bijective and homeomorphism.

Suppose $X$ be the set of all polynomial with real coefficients in one variable with norm $$\|p(x)\|=|a_0|+|a_1|+\dots+|a_n|$$ where $p(x)=a_0+a_1x+\dots+a_nx^n$ which induces a metric ...
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1answer
36 views

How to prove this statement?

I cannot prove this proposition directly . Let $(X,d)$ and $(Y,d')$ be metrice spaces. Let $f$ be a function from $X$ to $Y$. If $\overline{f^{-1} ( B)} \subseteq f^{-1}( \overline B)$ for all ...
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3answers
19 views

To show that $d $ and $ e$ are equivalent.

On the set $X = (0,1]$, consider the usual metric $d(x,y) = |x-y|, (x,y \in X) $ and another function $e: X\times X \to R$ given by $e(x,y) = |\frac{1}{x} - \frac{1}{y}|$. Show that $d $ and $ e$ ...
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3answers
36 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
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2answers
54 views

Showing that stereographic projection is a homeomorphism

For any $n\geq 0$,the unit $n$-sphere is the space $S^{n}\subset \mathbb{R^{n+1}}$ defined by $$S^{n}=S^{n}(1) :=\left\{ (x_{1},...,x_{n+1}) \left\vert\,\sum_{i=1}^{n+1} x_{i}^{2}=1\right.\right\}$$ ...
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1answer
28 views

about a sequence of isometries' convergency.

Let $M$ be a compact metric space, let $(i_n)$ be a sequence of isometries: $M \rightarrow M$. I've already showed that there exists a subsequence $(i_{n_k})$ that converges to $i$ which is also a ...
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1answer
19 views

Precompactness vs. Separability

I was always wondering... Given a metric space. To what extend do these notions differ: $$\Omega\text{ precompact}\implies\Omega\text{ separable}$$ (Precompactness meaning totally bounded.)
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1answer
40 views

Books or texts on singularity theory

So a friend is doing his PhD in maths (algebraic topology) and his advisor wants him to publish something on singularities (of which, as fas as I understand, he knows next to nothing). I want to give ...
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15 views

>If $A$ is an uncountable subset of a space whose topology has a countable base, then some point of $A$ is an accumulation point of $A$. [duplicate]

If $A$ is an uncountable subset of a space whose topology has a countable base, then some point of $A$ is an accumulation point of $A$. I've shown that there is an injection from A to the power ...
2
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1answer
36 views

Strange Quotient space $X / \mathbb{Z}$

For a practice-exam exercise I am trying to understand why $X/ \mathbb{Z}$ is homeomorphic to $S^1$. Here, $X = (-1,\infty)$, and $\mathbb{Z}$ is acting as an additive group on $X$ with the action: ...
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2answers
63 views

Series convergence and compact space

Let $K$ be a compact topological Hausdorff space. $\{x_n\}_1^\infty \subset K $ such that $x_i \not= x_j, i \not=j$ and $\{a_i\}_1^\infty \subset \mathbb{K}$. Show the folowing are equivalent: for ...
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0answers
57 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
3
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2answers
62 views

Finite groups and topological spaces

Can we connect topological spaces with groups as: For topological space $X$ take biective homomorfisms $\phi: X\to X$, then divide such homomorphisms on classes of equivalency $\phi_1 \equiv\phi_2$ ...
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1answer
43 views

Elementary Operations on Sets

Let $X$ be a set with subsets $A$ and $B$. Prove: a). $X \setminus (X \setminus A) =A$. $X \setminus A$ is the set of all points of $X$ which do not belong to $A$. Given $p \in X$, we will show that ...
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1answer
28 views

Fixed point of a map [on hold]

If g is a continuous map from U onto V in the complex plane, where U and V are homeomorphic to disks and U a proper subset of V. Must there be a fixed point? And if g is conformal, is this point ...
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0answers
6 views

Chain boundary (Topology)

Could anyone give me a topological definition of chain boundary, if possible one which could be integrated in further definitions (homology, quiver, bound chain and so on)??
4
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1answer
18 views

Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...