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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1answer
108 views

Foundational problem with set theory notation, and writing proofs out in the language of set theory

Let $E \subseteq \mathcal{P}(X)$ be a family of sets, then there exists a smallest toplogy containing $E$, i.e. $$ O_E := \bigcap_{E \subseteq O, O \textrm{ is topology}} O. $$ And now I read about a ...
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1answer
147 views

Intuition Regarding Homotopic Spaces

I am just starting to do some algebraic topology (very basic stuff) so have obviously just been introduced to the notion of homotopys, contractible spaces and homotopic spaces. It's this last one that ...
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2answers
132 views

Compact set mapped to compact set. [duplicate]

Possible Duplicate: Continuous images of compact sets are compact Theorem: Let V be a metric space. If $X\subseteq V$ and $f:X\rightarrow Y$ is continuous then $f(X)\subseteq Y$ is ...
6
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1answer
237 views

Euler characteristic fiber bundle

Let be $F \rightarrow E \rightarrow B$ be a fiber bundle where $B$ is path-connected and all three spaces are compact. Have you a simple proof that $\chi(E)=\chi(F) \chi(B)$? Where could I find that? ...
3
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2answers
116 views

Tychonoff cube $I^{m}$

Different references have different definitions for the Tychonoff cube. For example; R. Engelking, General Topology: The Tychonoff cube of weight $m‎‎‎\geq‎‎‎‎\aleph‎_{o}$ is the space $I^{m}$, ...
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3answers
489 views

What is your definition for neighborhood in topology?

As you know, Munkres-Topogy and Rudin-Analysis are really widely using textbooks for undergraduates. They all define a 'neighborhood of $x$' as an open set containing $x$, so i have followed this ...
2
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1answer
300 views

Subset of $C[0,1]$ is nowhere dense

Let $E_n$ be $$E_n:=\{f\in C[0,1]\mid \text{exist } x_f\in[0,1] \text{ such that } |f(x)-f(x_f)|\leq n|x-x_f|,\, \forall x\in[0,1]\}.$$ How show that $E_n$ is nowhere dense, that is, $\mathrm{int}\ ...
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3answers
346 views

Is every contractible space a cone?

It is easy to show that for any topological space $X$, the cone $CX$ is contractible. I am interested in the converse. If $Y$ is a contractible space, is $Y$ homeomorphic to $CX$ for some topological ...
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1answer
94 views

Using topology to show that a certain polynomial has a root

How exactly is it possible to use topology to show that every polynomial $f(x)= x^3 + ax^2 + bx +c$ has a root in $\mathbb{R}$? How about for a root in $\mathbb{C}$? I had no idea that topology had ...
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3answers
1k views

Return an array of evenly distributed points on a sphere give Radius and Origin. [duplicate]

Given a sphere of radius $r$, and origin $x,y,z$ what is the simplest way I can generate an evenly distributed array of points on the sphere $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots(x_n,y_n,z_n)$. Note I ...
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3answers
326 views

Help with metrics, Box topology and non-metrisable Hausdorff spaces

I'm trying to come up with a simple example of a second countable Hausdorff space that is not metrisable. The most promising I've come up with so far is the Box topology: The following is a basis for ...
2
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1answer
127 views

Is it necessary to have a normed space for the Heine-Borel-Property to hold?

The Heine-Borel-Property says: A subset M is compact iff it is closed and bounded. It is well known that the euclidean space $\mathbb{R}^n$ has this property. In a narrower sense I found the ...
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1answer
38 views

Let $\overline {X}_{0}\in \mathbb{R} ^{n}$ and $R>0$. Prove [closed]

Let $x_0\in\Bbb R^n$ and $R > 0$. Prove $\{x\in\Bbb R^n: \| x - x_0 \| \le R\}$ is complete.
4
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1answer
127 views

Cells decomposition of quotient space

Let $G$ a Compact Lie group and $T$ a maximal torus in $G$. I'd like to write an explicit Bruhat decomposition of $G/T$ and prove in this way that the Bruhat decomposition has $|W|$ cells of even ...
6
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1answer
636 views

Properties of the Mandelbrot set

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
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1answer
199 views

Zariski topology analogue for non-algebraically closed fields

Let $k$ be a field and $\bar{k}$ its algebraic closure. The set $X$ of $n$-tuples over $\bar{k}$ can be given the Zariski topology in which the closed sets are the sets of zeros of sets of polynomials ...
3
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1answer
88 views

Can a closed subset of an affine scheme have empty interior?

I have an inclusion of closed subsets $V(J) \subset V(I)$ in an affine scheme $Spec(R)$ with the property that $V(I) = V(J) \cup \partial V(I)$. I would like to conclude that $V(J)=V(I)$. (Here ...
2
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1answer
86 views

A question on linear ordered space

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace. My question ...
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2answers
471 views

Set with empty interior

What is the name of a set with empty interior? Wikipedia in a older version, say that is a hollow set, but i think that it is false. It is true? thanks in advance!
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1answer
513 views

Are there sets in the K-Topology that aren't open in the standard topology?

It seems to me that the basis for the K- Topology and the basis for the standard topology generate the same open sets. For instance, the open sets in the K-topology's basis that are different from ...
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1answer
138 views

Showing that the following statements are equivalent with regard to algebraic topology

Suppose we have a space $X$ that is path connected and locally path connected, and let $f: X \rightarrow Y$ be continuous. How do we show that the following statements are equivalent? (A) $f$ lifts ...
5
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3answers
252 views

Finding a topology on $ \mathbb{R}^2 $ such that the $x$-axis is dense

The problem is the following Put a topology on $ \mathbb{R}^2$ with the property that the line $\{(x,0):x\in \mathbb{R}\}$ is dense in $\mathbb{R}^2$ My attempt If (a,b) is in $R^2$, then define an ...
2
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1answer
81 views

How to apply product object in category to get product sigma algebra/topology/set systems?

Following is similar to my earlier questions, but try to understand them from category theory. Mariano said it was possible in a comment, but I don't know how. An object $X$ is the product of a ...
3
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1answer
1k views

Show that a retract of a Hausdorff space is closed.

A subspace $A \subset X$ is called a retract of $X$ if there is a map $r: X \rightarrow A$ such that $r(a) = a$ for all $a \in A$. (Such a map is called a retraction.) Proof. Let $x \notin A$ and $a ...
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1answer
416 views

What is difference between annulus (cylinder) and disk in graph routing?

What is difference between annulus (cylinder) and disk in graph routing? I know annulus is disk with hole, or I can imagine how is similar to cylinder, but my problem is, I can't understand this ...
2
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3answers
99 views

Closure of a subspace of $l^\infty$

Let $X$ be the following subspace of $l^\infty$: $$ X=\mathrm{lin}\{e_n:n\in\mathbb{Z}^+\} $$ where $e_j$ has zeroes everywhere except for one in the $j$-th entry. I want to know what the closure of ...
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1answer
161 views

Showing there is no retraction $r: B^2 \to S^1$.

For topological spaces $X \subset Y$ and a continuous retraction $r: Y \to X$, such that $\forall x \in X, r(x) = x$. How would you show, using the functoriality of the fundamental group, that there ...
2
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2answers
74 views

What $n$ s.t. this topology satisfies the separation axiom $T_n$?

Let $X$ be the real with the topology $T = \{ U \subset\mathbb{R}\:|\:0\notin U\text{ or }\mathbb{R}\setminus U\text{ finite}\} $. What is the largest value for $n \in \{0,1,2,3,3\frac{1}{2},4\}$ such ...
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1answer
59 views

Try to understand the definition of the property $D$ in one paper of J.E.Vaughan's

Try to understand the definition of the property $D$ in one paper of J.E.Vaughan's Firstly, let us see the Definitions(see page 238 of the paper): A countably infinite discrete set $A \subset ...
3
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2answers
192 views

If $X$ is normal and $A$ is a $F_{\sigma}$-set in $X$, then $A$ is normal. How could I prove this theorem?

A topological space $X$ is a normal space if, given any disjoint closed sets $E$ and $F$, there are open neighbourhoods $U$ of $E$ and $V$ of $F$ that are also disjoint. (Or more intuitively, this ...
2
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3answers
512 views

Every retraction is a quotient map?

I have to proof that every retraction is a quotient map.. I have no idea where to start or what to use! A retraction $r:X \rightarrow A$ is a continuous map s.t. $r(a)=a$ for every $a\in A$.
4
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136 views

finest non-discrete topology

Suppose we choose a topology which is not properly contained in any topology which is not discrete, appealing to Zorn's lemma. Does this yield anything interesting? I do not know a lot of topology, so ...
2
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2answers
200 views

Is this topology first countable?

We define the following topology on the reals; $T = \{ U \subset\mathbb{R}\:|\:0\notin U\text{ or }\mathbb{R}\setminus U\text{ finite}\} $. Is this topology first countable? I'm trying to find a local ...
14
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1answer
434 views

Metrics on $\mathbb R^{\mathbb N}$.

I was showing $D(x,y) = \sup_{k \in \mathbb N} \frac{d' (x_k,y_k)}{k}$ induces the product topology on $\mathbb R^{\mathbb N}$. Here $d(x,y)$ is the standard metric on $\mathbb R$ and $d'(x,y) = ...
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2answers
90 views

A question on continuity

In Munkres' "Topology", Section 18, Example 3 (pg. 104), it is stated that the identity function $$ g:\mathbb{R}_l\rightarrow\mathbb{R}, g(x)=x $$ where $\mathbb{R}$ has the usual topology and ...
5
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2answers
109 views

$f(\bigcap K_n)=\bigcap f(K_n)$? Where $K_n$ are compact

$f:X\rightarrow Y$ is continous map between metric spaces, $K_n$ are non empty nested sequence of compact subsets of $X$, then we need to show the title above. Please tell me which result I should ...
2
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1answer
580 views

Proof that Sorgenfrey plane is not normal using points x × (-x)

I'm making Exercise 9 of paragraph 31 in Munkres, which is a proof that the Sorgenfrey Plane $\mathbb{R}_l^2$ is not normal. I'm having trouble on part c of the question. The full question is: Let ...
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1answer
299 views

Is my understanding of product sigma algebra (or topology) correct?

Let $(E_i, \mathcal{B}_i)$ be measurable (or topological) spaces, where $i \in I$ is an index set, possibly infinite. Their product sigma algebra (or product topology) $\mathcal{B}$ on $E= \prod_{i ...
2
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1answer
119 views

Group of covering transformations for a universal cover

What would the universal cover of $S^1 \vee S^2$ be and what is its group of covering transformations? It's quite unfortunate that my topology class just very briefly touched on the subject which ...
3
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2answers
319 views

Function Spaces, why is the space of continuous functions (without necessarily differentiability) not important?

The space $C^0$ denotes the set of continuous and differentiable functions, the space $C^1$ the set of the continuous and differentiable functions which have a continuous and differentiable first ...
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2answers
108 views

Clopen quotient maps, does anyone know an example?

A quotient map is a surjection $\mathbb{f} : X \rightarrow Y$ such that; $O$ is open in Y $\Leftrightarrow$ $\mathbb{f}^{-1}(O)$ is open in X. This is an excercise in one of the later chapters of ...
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2answers
67 views

Does the finite intersection property apply here?

Suppose $X$ is a topological space. Now let $A_1 \supset A_2 \supset \cdots$ be a sequence of closed subsets of $X$. Suppose $a_i \in A_i$ $\forall i$ and $a_i \rightarrow b$. Prove that $b \in ...
3
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1answer
293 views

Cantor cubes are universal for totally disconnected compact Hausdorff spaces

Could any one tell me how to show that a totally disconnected compact Hausdorff space is homeomorphic to a closed subspace of a product of discrete two-point spaces? I am not able to see a known ...
3
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3answers
229 views

Passing to quotients via quotient maps preserving topological properties

Trying to review topology for a prelim, I'm starting to wonder exactly what topological properties do quotient maps, usually given as $p: X \rightarrow Y$, preserve? I believe quotient maps preserve ...
7
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2answers
862 views

Image of a normal space under a closed and continuous map is normal

$p : X \to Y$ is continuous closed and surjective, and $X$ is a normal space. Show $Y$ is normal. There is a hint, which I'm trying to prove: show that if $U$ is open in $X$ and $p^{-1}(\{y\}) ...
2
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0answers
106 views

Quotient map from the $(2n+1)$-dimensional sphere into complex projective space is open.

We have a natural quotient map $$\phi\colon S^{2n+1}\rightarrow (\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}=\mathbb{P}^n\mathbb{C}$$ and I want to see, that it's open. Denote by $$\iota\colon ...
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1answer
147 views

Is this subspace B lindelöf?

Let $X$ be the real numbers, and $r \in \mathbb{R}$ some real number. We define the following topology on X; $T = \{ U \subset\mathbb{R}\:|\:r\notin U\text{ or }\mathbb{R}\setminus U\text{ finite}\} ...
2
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2answers
370 views

Open Balls and Continuity

When we define continuity using open balls, we define $$\forall \epsilon >0, \exists\delta>0:f(B_\delta(a))\subset B_\epsilon(f(a))$$ Let us consider everything unspecified to be in ...
0
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1answer
95 views

Topology problem “2013 happy couples”

Let $$A=\{(x,y)\in \mathbb R^2\mid x^2+y^2\in \mathbb Z\},$$ $$G=\{(x,y)\in \mathbb R^2\mid y=|x|\},$$ $$B=\{(x,y)\in \mathbb R^2\mid d_2((x,y),G)\in\mathbb Z\}.$$ $f:A\to B$ is arbitrary continuous ...
1
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2answers
65 views

need to show $C[0,1]$ and $R^{\infty}$ is not localy compact

could any one tel me how to show the following? I am not getting any idea, thank you for help. $1)$ $C[0,1]$ is not locally compact. $2$) $\mathbb{R}^{\infty}$ is not locally compact where ...