Tagged Questions

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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0
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1answer
7 views

Showing a topological space covered by connected subspaces is connected

'Let $X$ be a topological space and let $(U_i)_{i \in I}$ be a cover of $X$ by connected subspaces $U_i$. Supposed for all $i,j \in I$ there exists some $n \geq 0$ and $k_0,...,k_n \in I$ such that ...
0
votes
1answer
14 views

Let $p: E \to B$ be a covering map. If $B$ is a completely regular space then prove that (edited) $E$ is completely regular space.

Let $p: E \to B$ be a covering map. If $B$ is a completely regular space then prove that $E$ is completely regular space. I am getting no clue how to construct the function $f$. The readers may ...
1
vote
1answer
11 views

Euler characteristic of closed surface

Assume that you have a closed surface that can be covered by finitely many triangles. Then $K(p)= 6-val(P)$ where P is a vertex and $val(P)$ the number of edges that lead to this vertex. Now, I am ...
1
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0answers
12 views

Cones over a topological space and homotopy!

Let $X$ be a topological space and let $CX$ be the cone over $X$. We identify $X$ with the subspace $X\times \{0\}$ of the cone through the immersion $x \mapsto [(x,0)]$ for $x \in X$. Let $f:X \to Y$ ...
1
vote
1answer
32 views

$f:X\rightarrow Y$ be a continuous bijection.$X$ and $Y$ are Banach spaces and $f$ is linear

Let $X$ and $Y$ be arbitrary topological spaces and let $f:X\rightarrow Y$ be a continuous bijection.$X$ and $Y$ are Banach spaces and $f$ is linear.To show $f$ is a homeomorphism.How to show $f$ ...
0
votes
0answers
21 views

The definiton of a discrete group

Is there a definition of a discrete group different from the one given in following link: http://en.wikipedia.org/wiki/Discrete_group
0
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2answers
28 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
0
votes
1answer
25 views

Show that a set is closed

I have to show that this set is closed: $\lbrace x \in \mathbb{R}^2 \vert \Vert x \Vert_2 \in [r,R] \rbrace$ Here $\Vert x \Vert_2 = \sqrt{x_1^2+x_2^2}$ and $R≥r>0$. I think that I have to use ...
0
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1answer
22 views

To Show that $S^n/(v\sim -v)$ is homeomorphic to $\mathbf RP^n$.

Let $S^n$ be the unit sphere in $\mathbf R^{n+1}$ and $\mathbf R P^n$ be the real projective space(see the definition of $\mathbf R P^n$ I am using in the References). Define a relation $\sim$ on ...
0
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1answer
15 views

Indiscrete space has trivial fundamental group [on hold]

How would you prove that any indiscrete space has trivial fundamental group.
0
votes
1answer
22 views

How to define open and closed functions whose domain or range is a discrete metric space?

I encountered that a function is open or closed in my analysis book [Herbert Amann, 2005], and it illustrates it in this way: A function $f: X \xrightarrow{} Y$ between metric spaces $(X,d)$ and ...
1
vote
1answer
19 views

Prove that the diagram $q: X \to Y$, $h \circ q^{-1}: Y \to Z$, $h: X \to Z$ commutes.

Suppose that the onto map $q: X \to Y$ is an identification, and $h: X \to Z$ is continuous. Assume $h \circ q^{-1}$ is single valued. Prove: 1) The function $h \circ q^{-1}: Y \to Z$ is continuous ...
0
votes
0answers
14 views

Net and filter generated by it

Let {s(a)} -such that (a) belongs to order set (A)- is a net from the point of (X) , the net {s(a)} converges to (x) if and only if the filter that generated by it converges to (x)
2
votes
2answers
43 views

$\mathbf R^2-\{\mathbf 0\}$ is homeomorphic to $S^1\times \mathbf R$.

I am trying to to prove the following: $\mathbf R^2-\{\mathbf 0\}$ is homeomorphic to $S^1\times \mathbf R$. Since $\mathbf R^+=\{x\in \mathbf R:x>0\}$ is homeomorphic to $\mathbf R$, it ...
2
votes
1answer
24 views

Why Not Define Connectedness to Mean Path Connected?

All spaces I have seen which are connected are also path connected (apart from examples to show that the two are not equivalent). Is there a reason for using the weaker definition of connectedness ...
1
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0answers
35 views

Lie group quotient structure

Let $G$ be a Lie group and $H$ a normal finite subgroup. Let $\pi : G \to G/H$ be the quotient surjection. How would one show that $G/H$ is a Lie group?
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0answers
25 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
0
votes
2answers
33 views

Help with some notation in QFT

I'm reading a paper on QFT and QEIs but i'm a little sketchy on some of the prerequisites. Can anyone tell me what this represents, $$C_{0}^{\infty}(M)$$ Where M is globally hyperbolic spacetime. I ...
1
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1answer
22 views

Closure of a left-bounded interval

I am trying to figure out what I might be doing wrong in this problem: Let there be a collection $\mathcal{A}$ of intervals such that $\mathcal{A} = \{(a, \infty) \mid a \in \mathbb{R}\} \cup ...
2
votes
1answer
29 views

Homeomorphic closed subspaces.

Let $X$ be an arbitrary topological space, and $U,V\subseteq X$ two subspaces of $X$ such that $U\cong V$ ($U$ and $V$ are homeomorphic) with respect the subspace topology of $X$. I know examples ...
0
votes
1answer
20 views

Baire's theorem and a set in $\mathbb{R}^n$

The following problem is definitely connected with a creative use of Baire's category theorem, but I didn't grasp the connection yet. We have $A\subset\mathbb{R}^n$ which is countable and a family of ...
2
votes
1answer
39 views

Build a topological manifold starting from a set.

Suppose you are given a generic set $X$. There exist sufficient and non-trivial conditions that ensure the existence of a topology $\tau_X$ on X such that the topological space $(X,\tau_X)$ is a ...
2
votes
1answer
33 views

Extenting a homeomorphism on subsurface to the entire surface

Suppose you have surface and a subsurface. The complement of subsurface is union of open discs and once punctured open discs. Can all homeomorphisms of the subsurface be extended to homeomorphisms of ...
1
vote
1answer
27 views

Homologous to zero but not contractible

Looking for instructive examples on the difference between homology and homotopy, I found here the following example: Example: Consider an oriented loop separating a genus $2$ surface into two ...
0
votes
2answers
23 views

Check if the parabola (with an induced topology) $\{(x,y)\in\mathbb{R}^2 | y=x^2\}$ is connected or compact.

i think yes connected but not compact, as it cannot be represented as a disjoint union and there is no finite sub cover. I'm just not sure how to go about proving this i.e. what to actually write ...
2
votes
1answer
45 views

Weak and weak* topology coincide for a non-reflexive space that is isomorpic to its dual?

There are Banach spaces which are isomorphic to their second dual but not reflexive (most famously, the James space). Now let $X$ be such a space and $X'$ be its dual space and let $\phi:X\to X''$ be ...
0
votes
1answer
24 views

Making an interval with point deleted complete

I am playing around with metric and topological spaces to get a better grasp of them, and I am wondering the following: is it possible to have a metric such that the set $[-1,0)\cup (0,1]$ is complete ...
2
votes
0answers
28 views

basis-free definition of linear function space topology

Let $V$ be a finite-dimensional space over $R$, and $M(V)$ the space of linear operators on $V$. I can choose an ordered basis in $V$ and identify it with $R^n$, and identify $M(V)$ with $R^{n^2}$, ...
1
vote
3answers
57 views

Convergence and topology

Please what is the classical method to answer this question, does the sequence converge in the given topology ? 1) The sequence $\big(1+(-1)^n\big)_{n\in\mathbb N}$ in $(\mathbb{R},\tau)$ such that ...
-2
votes
0answers
35 views

relation of these two continuity

Suppose that (1)$X=\mathbb{R}^n$. (2)$ M=\{ U$| $U$is mesurable subset of $X\}$. (3)$f:X\rightarrow X$ induces $f':M\rightarrow M$ s.t. $f'(U)=f(U)$ for all $U \in M$. (4)for $U\subset X$, $C(U)$ is ...
-1
votes
1answer
51 views

Show these sets are homeomorphic to eachother

1) {${(x, y) ∈ R^2 |0 < x^2 + y^2 < 1}$} 2) {${(x, y) ∈ R^2 | x^2 + y^2 > 1}$} I've considered mapping r to 1/r, from (0,1) to (1,∞)
-1
votes
0answers
5 views

About Hausdorff characterization

I am thinking about why in a complete metric space, A is a compact subset <=> A is totally bounded and close. What about in an incomplete metric space? Anyone can help me? Thanks a lot!
0
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0answers
27 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
-1
votes
0answers
12 views

Semi open Sets in topological space

As union of two semi open sets is semi open and intersection may not be semi open. whether sum of two semi open sets is semi open or not?
2
votes
0answers
15 views

When is a knot complement a solid torus?

Let $M$ be an oriented 3-manifold. If $K \subset M$ is a knot whose complement $M \setminus N(K)$ deform retracts onto a circle, is $M \setminus N(K)$ a solid torus? Here $N(K)$ is an open normal ...
0
votes
1answer
25 views

What is the support of the Dirichlet Function?

The Dirichlet function is defined as: if $x$ is rational, $f(x)=0$, if $x$ is irrational, $f(x)=1$. What is its support? I think the answer is the set of irrational numbers.
1
vote
1answer
33 views

Why is it hard to prove Jordan Curve Theorem in the case of Koch snowflake

Many books and papers mentioned that it is easier to prove Jordan Curve Theorem in the case of polygon and hard in the case of badly behaving curves. One example that most give is Koch snowflake. My ...
1
vote
1answer
37 views

Orange Peeling Problem

Given any sphere, in this case, the most suitable sphere would be an orange, could you "peel" of an orange such that when it's fully peeled, the orange peel is a genus-1 loop? Is there any conjectures ...
0
votes
0answers
34 views

Map between manifolds.

Let $M,N \subset \mathbb{R}^3$ be (not necessarily smooth) 2-manifolds without boundary. Let $f: M \rightarrow N$ be a continuous function and suppose that $f$ is injective. Let $x \in M$ and let $U$ ...
1
vote
1answer
36 views

An alternative proof of the Tietze Extension Theorem(s)

Last summer I was working through a lot of Topology. I made it through the sections of my notes that dealt with separation properties, covering properties and continuous functions between spaces ...
0
votes
0answers
42 views

Divisibility lattice and duality with topological spaces

Consider the integers $\mathbb{N}$ seen as a poset with divisibility as an order relation. See it as a distributive bounded lattice with gcd and lcm, with gcd being the meet and lcm the join. Clearly, ...
0
votes
1answer
51 views

Only one fixed point for $f:\bar{\mathbb{D}}\rightarrow\bar{\mathbb{D}}$ on the boundary.

We know for Brouwer theorem that $f$ (continuous bijective function) have a fixed point. My questions are: 1) Is there a function with only one fixed point $x_0\in Int(\bar{\mathbb{D}}) $ (open ...
-2
votes
0answers
20 views

Acceptance-Rejection Method [on hold]

Consider the PDF of a random variable $X$ defined as follows: $$ f(x) = \begin{cases} x(1-x/2) & \text{ if $0 \leq x<1$} \\ 0 & \text{ otherwise} \\ \end{cases} $$ Using ...
0
votes
1answer
17 views

Show that if $h$ is extendable to a continuous map of $\Bbb R^n$ into $Y$, then $h_*$ is the trivial homomorphism.

Let $A$ be a subspace of $\Bbb R^n$; let $h:(A,a_0) \to (Y,y_0)$. Show that if $h$ is extendable to a continuous map of $\Bbb R^n$ into $Y$, then $h_*$ is the trivial homomorphism. I can't get any ...
2
votes
2answers
48 views

Example of Hausdorff and second countable space which is not metrizable

Does there exist topological space which is Hausdorff and second countable but which is not metrizable?
1
vote
1answer
20 views

Homeomorphism from $O(n)/(O(1)\times O(n-1))$ to $S^{n-1}/\tilde{}$

I'm trying to find a homeomorphism from $O(n)/(O(1)\times{}O(n-1))$ with $O(1)\times{}O(n-1)$ being Matrizes with 1 or -1 in the top left and a matrix of $O(n-1)$ at the bottom right to the Sphere ...
1
vote
0answers
19 views

Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
1
vote
0answers
36 views
+100

Approximation of holomorphic functions and topological properties

So, in the last couple of lectures of my complex analysis class we've proved some approximation theorems of holomorphic functions. Eventually, we showed the following propositions: Theorem 1. Let ...
1
vote
3answers
19 views

Construction of a compact set with 3 limit points

This problem is asked in this website, but I have some confusion about it still. I need to construct a compact set with exactly 3 limit points the set $A={\frac{1}{n}}: n\in \Bbb Z$ has only one ...