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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A question about general topology and convexity in Euclidean spaces

Let m be a positive integer and let E(m) be m-dimensional Euclidean space with its standard metric. For any positive integer n greater than m, let P(1),P(2),...,P(n) be a finite set of points of E(m) ...
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1answer
29 views

Are these statements about wedge sum true?

Since there's no concrete explanation on wedge sum in my text, I have proven following basic statements on my own. I want to know whether I proved correct things. First of all, here is the definition ...
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0answers
17 views

Show that the quotient map G-> G/H is a covering space [duplicate]

G be a topological group.H be a subgroup of H.suppose that the subspace topology on H is the discrete topology.Show that the quotient map G-> G/H is a covering space. Prove that the quotient map $P:G ...
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4answers
61 views

Show that $A=\{x\in X\mid a\leq f(x)\leq b\:;\;a,b\in\mathbb{R}\}$ is closed if $f:X\to \mathbb R$ is continuous.

Let $X$ be a set. Suppose that $f:X\to\mathbb{R}$ is a continuous function and let $A=\{x\in X\mid a\leq f(x)\leq b\:;\;a,b\in\mathbb{R}\}$. Is $A$ closed, open, clopen or none? So I started by ...
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34 views

How do I prove that this map is a homeomorphism?

Let $X$ be a topological space. Let $\{X_i\}$ be a family of mutually disjoint open subsets of $X$ such that $\bigcup X_i = X$. Let $a_i$ be a point of $X_i$ for each $i$. Consider a quotient map ...
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1answer
32 views

Completely regular space with a G$_\delta$-singleton which is not a zero-set

Is there a completely regular (Hausdorff) space in which all singleton subsets are Gδ but which has a singleton subset which is not a zero-set? (Better yet, a first-countable such space.) ...
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3answers
70 views

The set of accumulation points of A is closed

I'm having a bit of a hard time proving or disproving the following claim in general topology: Let X be a topological space, A $\subseteq$ X, and B the set of accumulation points of A. Is B ...
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27 views

Metrizable topological space

Why Extended real numbers set with T ( T topology on R with infinity) , is metrizable ? And how can prove that d(x,y) genetares this topology (T) ??
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16 views

Show that folowings spaces are homeomorphics [closed]

The closed ball in $R^2$ centered at $0$, radius $1$ and the closed square $[0,1]\times [0,1]$ thanks
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1answer
41 views

Is every continuous one-to-one image of $[0,\infty)$ locally compact?

Suppose $f:[0,\infty)\to Y$ is continuous and one-to-one onto $Y$. You may assume $Y$ is metric. Is $Y$ locally compact? Thanks!
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34 views

Cantor Space - Example - Proving Compactness/Perfectness/Closed/Totally disconnected

Say we take the set of infinite binary codes $\{0,1\}^\mathbb{N}$, which is often written as $2^\mathbb{N}$, mapped to the Cantor set defined previously as $C_n=\frac{c_{n-1}}{3} \cup \left( ...
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1answer
50 views

Zero sets in Mrówka spaces

For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in ...
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3answers
90 views

Why is the middle third Cantor set written as this?

My first question is, is the middle third Cantor set the same as the Cantor set? I've never heard it called the middle third Cantor set. Secondly, why is this true: "I’m going to assume that Cantor ...
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54 views

Cantor Space - Example

Consider the map from the set of infinite binary codes $$\{0,1\}^\mathbb{N}$$often written $$2^\mathbb{N}$$ to the Cantor set defined above: $f:2^\mathbb{N}\to C$ defined as for a sequence: ...
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1answer
32 views

Removing a open set from a finite open covering of a Normal space.

Like the title says: ¿The subspace resulting by removing a element of a finite open covering of a normal space $X$, is also normal space? In symbols: Let $(X,\tau)$ be a normal space and $\{U_1, U_2, ...
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1answer
81 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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0answers
38 views

Understanding the mechanics of P-adic topologies

I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open ...
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2answers
23 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 ...
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1answer
46 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 ...
2
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1answer
37 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 ...
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1answer
36 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 $f$ ...
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0answers
28 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
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2answers
38 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 ...
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1answer
28 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 ...
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1answer
25 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 ...
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1answer
22 views

Indiscrete space has trivial fundamental group [closed]

How would you prove that any indiscrete space has trivial fundamental group.
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1answer
29 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 ...
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1answer
23 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 ...
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1answer
21 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)
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$\mathbf R^2-\{\mathbf 0\}$ 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 ...
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1answer
29 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 ...
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0answers
47 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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27 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 ...
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2answers
34 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 ...
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1answer
25 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
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1answer
30 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 ...
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1answer
25 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
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1answer
49 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
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1answer
42 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 ...
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1answer
33 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 ...
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2answers
24 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
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1answer
50 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 ...
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1answer
28 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 ...
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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 $R^{n^2}$, ...
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3answers
75 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 ...
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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 ...
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1answer
56 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,∞)
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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!
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33 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 ...
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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?