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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2answers
52 views

Strong version of Baire Category Theorem

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense. In such spaces is the intersection of fewer than $2^\omega$-many open ...
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0answers
19 views

Hilbert cube in metric Spaces

Is there proof that does not use the concept of " product space " to prove that the Hilbert cube is closed and bounded ?
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0answers
18 views

A problem of a collection of subsets maximal with respect to the finite intersection property [duplicate]

Let $X$ be a space. Let $\mathscr {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that if $X$ is Hausdorff, there is at most one point ...
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0answers
37 views

If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y ...
4
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1answer
31 views

connected space of matrices

Let $M(n, \mathbb{R})$ be a set of square matrices. Consider the subset $S$ of $M(n, \mathbb{R})$ where the absolute value of the eigenvalues of the matrices in the subset are $\le 2$. Is this subset ...
2
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2answers
42 views

A problem in topology relating to the finite intersection property

This is a problem from Munkres' Topology Exercise 37.1 (c) Let $X$ be a space. Let $\mathscr{D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. ...
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3answers
60 views

Complete metric space, not simply-connected

I've been going over the algebraic topology part of Munkres and this question has stumped me. If we have a complete metric space that is not compact, must it be simply-connected (path-connected plus ...
2
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0answers
30 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
2
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1answer
43 views

Counterexamples to Brouwer’s fixed point theorem

Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say ...
1
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1answer
30 views

Show that for a degree 1 map $f: M \rightarrow N$ the induced map $f_*: H_1(M) \rightarrow H_1(N)$ is a surjection

I'm trying to solve the following problem: Show that for a degree 1 map $f: M \rightarrow N$ of connected, closed, orientable manifolds, the induced map $f_*: \pi_1(M) \rightarrow \pi_1(N)$ is ...
3
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2answers
63 views

Most general definition of homomorphism and isomorphism

What is the most general definition of homomorphism and isomorphism? It is clear what they mean when there is an algebraic structure to be preserved, but what about when there is no such structure? ...
1
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1answer
23 views

A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$

This is a theorem from Topology by James Munkres. Theorem 34.3 A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$ The book merely states that ...
5
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1answer
86 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
0
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1answer
42 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
1
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1answer
27 views

Kuratowski's definition of a topological space, and immediate consequences

According to Kuratowski, a topological space is a set $X$ together with an operation ${\bf C}$ (called closure) which associates to every element $A\subset X$ an element ${\bf C}A\subset X$ such that: ...
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2answers
27 views

Is $(-3n,3n)$ a subcover of $(-n,n)$?

I read this: Let $C=\{(-n,n):n \in \mathbb N\}$ and $C'=\{(-3n,3n):n \in \mathbb N\}$. Then $C$ and $C'$ are open covers of $\mathbb R$ (understood) and $C'$ is subcover of $C$ (not understood). A ...
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1answer
12 views

Does the Function Between Some Set and N Has Any Properties? (Question about Countable Set)

(Note that when I say countable, it means countable infinite) My question is that, the definition of a countable set is $S$ is said to be a countable set if there is a bijective function ...
2
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0answers
49 views

Criterion for orientability: Derivative of transition map

The definition I have been given for a smooth abstract surface, $S$, to be orientable is that given a continuous family of maps $f_t: D \to S$ that embed the closed unit disk into $S$ with $f_0(D) = ...
2
votes
1answer
22 views

Is a closed map from X to T1 proper?

$f:X→Y$ is a closed map. Suppose that the inverse image of each point in $Y$ is a compact subset of $X.$ Show that $f^{-1}(K)$ is compact in $X$ whenever $K$ is compact in $Y.$ This is my homework ...
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1answer
41 views

Nature of Equilibrium Points

I would like to prove the following: "The nature of the equilibrium points (i.e. stability/instability) of a one-dimensional differential equation remains invariant under the effect of the ...
0
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1answer
39 views

How to build a covering space ?? [closed]

I have to build a covering space $ p:\mathbb{R} ^{2} \rightarrow S^{1} \times S^{1}$ How to do it?
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0answers
26 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
0
votes
1answer
31 views

limit points of $[0,1]$

$[0,1]$ defined in $\mathbb{R}$ is a closed set. A closed set contains all its limit points. The points $0$ and $1$ do not have all their neighbourhoods within $[0,1]$. So they are not limit points ...
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1answer
85 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
0
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1answer
20 views

Prove a given set on the euclidean plane is connected

Let $f:\mathbb{R}\longrightarrow\mathbb{R},\ g:\mathbb{R}\longrightarrow\mathbb{R}$ be functions, that satisfy $f(x)\leq g(x)\ \forall x\in\mathbb{R}$ and let $S=\{(x,y)\in\mathbb{R}^2:f(x)\leq y\leq ...
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4answers
59 views

Is a continuous bijection function from a hausdorff space to a compact space is a homeomorphism?

We know a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism. But I am wondering what happened if we switch the domain and codomain. Is a continuous bijection ...
2
votes
1answer
49 views

A question (more like three) about a topological space of ordinals.

I've been struggling with these for a while now, if anyone is willing to offer a hint I'll be more than grateful. Given an ordinal $\varepsilon$, consider the topological space $L_{\varepsilon}$ ...
3
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0answers
44 views

Is the distance induced by the topology of the set a metric

First of all excuse me if something with the question is wrong. I am not very knowladbable in this area, though I know what I am asking and my question should be infact a reasonable one. The topology ...
1
vote
1answer
41 views

How to draw a quotient space

I'm trying to understand this exercise: What does the author mean? I could find proprieties of this quotient space and everything but I don't know how to draw it, besides that it seems there are a ...
1
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1answer
36 views

A metric and discrete topology

Let $\Sigma=\{1,2,...,n\}$ and $\Omega=\Sigma^\mathbb{N}$ be the set of infinite sequence of n digits. Define a metric $d$ on $\Omega$ by $d(\omega,\tau)=2^{-|\omega\wedge\tau|}$ where ...
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0answers
17 views

The first countable space [duplicate]

Let X uncountable set and let (X,Tcf) is a topological space such that Tcf: I mean complement finite topology why this topological spaces isn't first countable space ?
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0answers
25 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
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0answers
22 views

$X$ is a topological space of infinite cardinality which is homeomorphic to $X × X$. [duplicate]

$X$ is a topological space of infinite cardinality which is homeomorphic to $X × X$.Then which is true A. X is not connected B. X is not compact C. X is not homemorphic to a subset of R D. none of the ...
0
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2answers
49 views

How to show the covering space of an orientable manifold is orientable

I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it. I've been playing around with definitions a bit and here's what I have so far. ...
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0answers
55 views

Number of connected components of $f^{-1} (U)$

Let $f:\mathbb{R}^n \to \mathbb{R}$ be an analytical function (semialgebraic,polynomial if needed), $U$ be an open connected subset of $\mathbb{R}$. What can we say about the nuber of connected ...
0
votes
1answer
53 views

Isomorphism from a subset of $\mathbb{R}^n$ to the space $\mathbb{R}^{n-1}$

Let $I \subset \mathbb{R}^n$ be non-empty such that $I$ is in general not a vector subspace of $\mathbb{R}^n$, i.e. it is not necessarily closed under addition and multiplication by a scalar. We ...
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2answers
38 views

Which of the following are true about homeomorphic

Let $X=[-1,1]×[-1,1], A=\{(x,y)∈X: x^2+y^2 =1\}, B=\{(x,y)∈X:|x|+|y|=1\}$ $C=\{(x,y)∈X: xy=0\}$ and $D=\{(x,y)∈X: x=±y\}$.Then A. A is homeomorphic to B B. B is homeomorphic to C C. C is ...
1
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2answers
32 views

Closed Set in Relative Topology

I am pretty sure this result is true but I am feeling a little bit lazy trying to prove it rigorously (done way too many point-set proofs). Does anyone have a very short and elegant proof without too ...
2
votes
3answers
101 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
0
votes
1answer
22 views

Open or closed in relative metric

$C=(1/4, 3/4)$, is $C$ open in $\mathbb{R}^2$? I think it is open by definition of open. But I am not sure what is the difference between $C$ is open in $\mathbb{R}$ and $C$ is open in $\mathbb{R}^2$. ...
0
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0answers
31 views

Induced topology by a complete uniform space.

I know that Uniform space is generalization idea of metric space,Uniform space like metric space induce a topological space. Now my question is ( or are ):- In case our Uniform space was complete ...
2
votes
1answer
28 views

Operator topologies and examples

In class we covered several operator topologies: the weak topology, the weak* topology, the weak operator topology, and the strong operator topology. The first two are defined on a normed vector ...
2
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0answers
15 views

Double adjoints and reflexivity

Let $X$ and $Y$ be normed (or Banach) spaces. Does anyone know a nice proof that every bounded operator $T:X \rightarrow Y$ is its own double adjoint (that is $T^{\ast\ast}=T$) if and only if $X$ and ...
0
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1answer
28 views

Confused about open set

This statement appears in book : {0} is not open in R. But according to the definition of open, I can find an open ball centered at 0 lies in R, should it be open?
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2answers
25 views

M is a metric space, then M is open

If M is a metric space, how to prove that M is open by itself. Let x in M , and there exists an open ball centered at x lies in M. How to reason this statement?
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1answer
20 views

Closed function and adherence

I have by definition that a function is cloded if the image of a closed set by this function still closed And i want to prove that $$f:E\rightarrow F ~\text{is closed} \Leftrightarrow ...
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0answers
33 views

Can you deformation retract a sphere to a point?

So, I'm working on a topology problem (Calculating the fundemental group of two spheres adjoined by a single point). As a subpart of the problem, we're trying to figure out if a sphere by itself can ...
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0answers
24 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
0
votes
1answer
29 views

Normal space and discrete family

If $X$ is a normal space and $\{ F_\alpha : \alpha \in A \}$ is a countable discrete family of closed sets in $X$, then how to prove it that we can find a discrete family $\{O_\alpha : \alpha \in A ...
2
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0answers
22 views

Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...