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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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0answers
21 views

A function from the $S^1$ to $S^1$ homotopic to the constant map?

How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$ (a) has a constant point and that (b) $f$ maps $x$ to its antipodal point $-x$?
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0answers
6 views

$\phi$ is a coboundary iff $\phi(f)$ depends only on the endpoints of $f$

I've proved the first direction but I'm having trouble proving the second direction. First direction: Let $\phi = \delta\psi$ for $\psi \in C^0(X;G)$. Then $\phi(f) = \delta\psi(f) = ...
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1answer
22 views

How to prove a map between two spaces of real sequences $f : l^1 \to l^2 $ is well-defined and continuous

the question is whether the following statement is ture or false, and justify it. Here is the statement The map $f : l^1\to l^2$ given by $f(x_0, x_1, x_2,...)= (x_0, x_1, x_2,...) $ is ...
3
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2answers
92 views

A continuous function from the open ball to itself?

How to prove that there exists a continuous function $f:B^2 \to B^2$ without constant points? Here, $B^2$ is the unit open ball. I guess $f$ can be for example like this $f: re^{iax} \to re^{ibx} $ ...
2
votes
1answer
33 views

A subset that is not open

how can I show that if $x$ and $y$ are two distinct points in a Hausdorff space with $U$ and $V$ the neighborhoods of $x$ and $y$ respectively, then $( U$ \ {$ x $} $)$ $\cup$ {$y$} is not open. ...
1
vote
1answer
37 views

Proof of Jordan curve theorem

Is it possible for the following to be proof for Jordan curve theorem: Given the distance function on $\mathbb{R}^2$ ($d((x_1,y_1),(x_2,y_2))=\sqrt{ |x_2-x_1|^2 + |y_2-y_1|^2}$), and $\varepsilon ...
0
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0answers
9 views

Is this statement true?(covering map)

Let $C,X$ be topological spaces. Let $p:C\rightarrow X$ be a continuous function. Let $U$ be an evenly covered open subset of $X$. Let $V$ be an open subset of $C$ such that ...
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1answer
14 views

not metrizable?

In Munkres, section 30, exercise 6 is this: Show that $R_{l}$ and $I^2_0$ are not metrizable. I guess $R_{l}$ is lower limit topology, and $I^2_0$ is an ordered square. and here, how to prove they ...
1
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0answers
24 views

Showing $\phi(f \cdot g) = \phi(f) + \phi(g)$

For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$. What I'm not sure is how I'm supposed to relate a ...
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0answers
12 views

Question on weak star convergence in subspace

Let $X$ be some normed linear space and let $X^\ast$ denote its dual space endowed with the weak star topology. Let $U^\ast$ be some subspace of $X^\ast$. If I want to show that $\varphi_n$ ...
2
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0answers
31 views

Does there exist a continuous function between the following sets:

Does there exist a continuous function between the following sets: $A.f:(-1,1)\rightarrow (-1,1]$ which is onto and one-one $B.f:\{(x,y):y^2=4x\}\rightarrow \mathbb R$ which is one-one What ...
0
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0answers
28 views

Every map from a compact Hausdorff space is continuous

Could someone help me figure out my mistake? I just proved that if $X$ is compact Hausdorff then $f: X \to Z$ is continuous. Here's my proof: Let $U$ be open in $Z$. Let $x \in f^{-1}(U)$. Since $X$ ...
2
votes
1answer
31 views

Equivalence classes of $\mathbb R$

Let $X$ be a locally compact, connected, locally connected, Hausdorff space. Considder $U_1\supseteq U_2\supseteq\cdots$ of open and non-empty connected subsets with compact frontiers such that ...
0
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1answer
27 views

$\{(x,y)\in \mathbb R^2:xy=1\}$

To check which pairs are Homeomorphic? A.$\{(x,y)\in \mathbb R^2:xy=0\}$ B.$\{(x,y)\in \mathbb R^2:xy=1\}$ C.$\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$ D.$\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}$ I ...
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0answers
43 views

Is the Zariski Topology

if $ K $ is an algebraically closed field, asks: Is there a point $ "w" $ of $ K ^ n $, is closed in the Zariski toplogy?
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1answer
33 views

Open map in complex analysis

Could anyone please help me with the following problem, Let $f(z)={\rm Re}(z)$, then show that $f:\Bbb C\to \Bbb R$ is an open map but not a closed map whereas $f:\Bbb C\to \Bbb C$ is neither an open ...
0
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1answer
16 views

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) ...
0
votes
1answer
28 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 ...
-1
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0answers
16 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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vote
4answers
59 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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
30 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.) ...
2
votes
3answers
64 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 ...
0
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0answers
24 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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0answers
14 views

Show that folowings spaces are homeomorphics [on hold]

The closed ball in $R^2$ centered at $0$, radius $1$ and the closed square $[0,1]\times [0,1]$ thanks
0
votes
1answer
39 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!
0
votes
0answers
30 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( ...
1
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1answer
38 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 ...
2
votes
3answers
88 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 ...
0
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0answers
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: ...
0
votes
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, ...
2
votes
1answer
78 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 ...
1
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0answers
33 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 ...
0
votes
2answers
21 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 ...
1
vote
1answer
44 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
votes
1answer
28 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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0answers
27 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
35 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
25 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 ...
0
votes
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 ...
0
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1answer
24 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
18 views

Indiscrete space has trivial fundamental group [on hold]

How would you prove that any indiscrete space has trivial fundamental group.
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1answer
27 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
22 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
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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vote
2answers
44 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
27 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
44 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
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 ...