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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2answers
24 views

a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$

Prove that a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$. The direction $\Rightarrow$ was easy. But I don't ...
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1answer
52 views

can anyone help me with following question attached in image file [on hold]

Let $(X,\|\cdot\|)$ be a normed space, where $$X=\{(a_n)_{n\geq 1} \mid (a_n)_{n\geq 1} \text{, bounded real sequence}\}$$ and $$\|(a_n)_n\|=\sup_{n\in N} |a_n|$$ Let $$ M=\{(a_n)_n\in X\mid 0\leq ...
3
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0answers
40 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
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1answer
21 views

Separable iff Lindelof for pseudometric spaces

I'm trying to prove, for $X$ a pseudometric space $$X \text{ Lindelof } \Leftrightarrow X \text{ separable }$$ Here are some of my ideas so far - the forward direction should work: $(\Rightarrow)$ ...
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1answer
16 views

How to prove that every Paracompact space with the Suslin property is Lindelof

This question was asked a few years ago and a proof was given here http://math.stackexchange.com/a/190147/235467. However, in this proof it states that paracompactness implies the existence of a ...
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4answers
36 views

$\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$

Why is $\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$ (here is $\mathbb{R}$ endowed with the standard topology? I could use the criterion: Is $(x_n)\subseteq M$ such that $x_n\to ...
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1answer
14 views

$\partial M\subset M$ implies (Is $(x_n)\subseteq M$ such that $x_n\to x_0\in\mathbb{R}^n \Rightarrow x_0\in M$)

Let $M\subset \mathbb{R}^n$. I want to how to proof: Why implies 1. $\partial M\subset M$ this type of closedness: 2. Is $(x_n)\subseteq M$ such that $x_n\to x_0\in\mathbb{R}^n \Rightarrow x_0\in M$? ...
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0answers
31 views

Set of continuous functions that vanish at infinity is complete

Why is it easy to see that a set of all continuous functions $C_0$ that vanish at infinity implies that each $f\in C_0$ is bounded and the set is complete with respect to the uniform (sup) -norm? ...
2
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1answer
40 views

Open and Closed covering

Let $X$ be a compact Hausdorff and totally disconnected space and $A$ be a closed subset of $X$ contained in an open set $U$. Then we can find a finite set $\{V_1,\cdots,V_n\}$, where each $V_i$ is ...
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2answers
18 views

Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product ...
2
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0answers
37 views

Tetrahedron and balls in space

A right tetrahedron and a ball arbitrarily located in space are given. It is allowed to reflect the tetrahedron from each of its faces. It is possible to place the center of the tetrahedron inside the ...
0
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1answer
26 views

Topology on $[X]^2$ for Hausdorff space $X$

Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in ...
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1answer
20 views

Proof product of components in factors is a component in product topology

Let $x = (x_1, x_2, .... x_{n})$ be a point in a product space $(Y, \tau_{Y}) = \prod_{i = 1}^{n} (X_{i}, \tau_{i})$. The component $C_{X}(y)$ in a topological space is the union of all connected ...
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1answer
43 views

If there is a finite-closed topology on $X$ with 3 clopen elements, then $X$ is finite

Let $T$ be a finite-closed topology on $X$. $X$ has 3 clopen elements. Prove that $X$ is finite. Empty set must be one of these clopen sets as well as $X$. Therefore, we are left with some element ...
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1answer
68 views

Covering spaces of $S^1$

Put $\tilde X=\lbrace (exp(2\pi if(t)),t)| t\in \mathbb{R} \rbrace$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is any continuous function and let $\pi_1$ be the projecction on the first coordinate. ...
2
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3answers
51 views

(Non-Euclidean) Compactness

Compactness in Euclidean Space The only definition of compact set that ever made sense to me was the intro calculus one: A set is called compact if it is closed and bounded. ...
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1answer
54 views

The intersection of dense subset and open subset

Let $A$ be a dense subset of $X$, and $B$ let be a non-empty open subset of $X$. Prove that $A\cap B \not = \emptyset $. if A is dense in X then $ \bar{A}=X=A\cup A'$ where $A'$ is the derived set ...
0
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0answers
18 views

Relationships Between Moduli Space and Objects They Parametrize

My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the geometry of the objects that the space parametrizes. As an ...
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1answer
43 views

Question on one point compactification

I was given the following question in my general topology class assignment which is multi parts - most of which I managed alright by myself some of which I need help on. We are given a non compact ...
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1answer
98 views

Can something contain iteself? [on hold]

I asked this over on the Phyisics part of StackExchange, and they suggested I move my question here. And said question is: Can something contain itself? The question is simple enough, and I can ...
3
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4answers
94 views

Showing $\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace$ is Closed

Let $K=\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace \subseteq \mathbb{R}^2$. Show that $K$ is closed. I am following Munkres' topology book, and this is a step towards finishing problem 3 on p. ...
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1answer
25 views

prove finite intersection property for compact sets using sequential compactness

Prove finite intersection property for compact sets in metric spaces using sequential compactness with a direct proof . One approach is to prove sequential compactness and covering compactness are ...
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0answers
24 views

converging subsequences of two metrics

if $d$ and $d'$ are two metrics on a space $X$, is it true that they induce the same topology if and only if they have the same converging sequences ?
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1answer
39 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
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0answers
15 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
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3answers
62 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
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1answer
35 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
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2answers
51 views

Show that $A$ is open in $\mathbb R$

I got this question in a test earlier today. I know it is a very small question, since it only counted 2 marks, but for some reason I simply could not get it?? Let $f:\mathbb R \to \mathbb R$ be ...
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1answer
45 views

Compact subsets of the space of real functions $\mathbb{R}^\mathbb{R}$

I was suprised that this question hasn't been asked - or maybe it was, but asked differently. Anyway, I want to characterize the compact sets in the space of real functions $\mathbb{R}^\mathbb{R}$ ...
4
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2answers
199 views

Space which is neither locally connected at any point nor totally disconnected

Let $X$ be a topological space; then we say that $X$ is locally connected at $x$ if $x$ admits a neighborhood basis of open, connected sets. In this sense, a space is locally connected iff it is ...
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0answers
47 views

Show that if $f$ is a proper surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is ...
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0answers
34 views

Finite subset of $\Bbb R$ is nowhere dense [on hold]

I need to show that every finite subset of $\Bbb R $ is nowhere dense. Thanks
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0answers
27 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
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1answer
47 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
3
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1answer
33 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
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4answers
56 views

What is induced topology?

In my text, it says "Given a topological space $X$ and a subspace $S ⊂ X$, define the induced topology on $S$ to be the topology in which the open sets are of form $U ∩ S$, where $U$ is open ...
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1answer
22 views

Product (arbitrary) of open functions is open.

Let $f_{\alpha}\colon X_{\alpha}\to Y_{\alpha}$ be open, for all $\alpha \in J$. Then $\prod_{\alpha} f_{\alpha}\colon \prod_{\alpha}X_{\alpha} \to \prod_{\alpha}Y_{\alpha}$ is open? Both $ ...
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1answer
41 views

Box topology and axiom of choice

Below is the definition of box topology: Given an indexed family of topological spaces $X_\alpha $, the collection of all sets of the form $$\prod_{\alpha\in J} U_\alpha,$$ where $U_\alpha$ is open ...
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0answers
22 views

It is correct this definition of limit of a function?

I have a definition of the limit of a function in some point $\alpha$ for metric spaces on this manner: We have two metric spaces $(E,d)$ and $(F,p)$; $A\subset E$, $f:A\to F$. Then ...
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1answer
74 views

Unifying Connection Between Topological Embeddings and Quotient Maps

In a book on topology I'm reading the following theorem seemed striking to me, not for its proof, which I believe I understand, but because there's some nice symmetry going on that I'd perhaps like ...
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1answer
42 views

An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
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1answer
57 views

Necessarily a homeomorphism?

Let $D$ be the projective curve defined by $y^2z = x^3.$ Consider the map $f: \mathbb{P}_1 \to D$ defined by$$f[s, t] = [s^2t, s^3, t^3].$$Is it necessarily a homeomorphism? Any help would be greatly ...
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1answer
21 views

General Form of a Open Set in the Product Topology in a Countably Infinite Product.

Suppose $\{X_n\}_{n\in\Bbb N^+}$ is a family of topological spaces. I understand that a typical basis element of the product topology has the form $$\prod_{n=1}^k U_n\times\prod_{n=k+1}^\infty ...
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0answers
44 views

The degree-genus formula cannot be applied to singular curves in $\mathbb{P}_2$?

(The degree-genus formula) The Euler number $\chi$ and genus $g$ of a nonsingular projective curve of degree $d$ in $\mathbb{P}_2$ are given by$$\chi = d(3-d)$$and$$g = {1\over2}(d-1)(d-2).$$ My ...
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0answers
23 views

Which definition of a neighborhood is more standard? [duplicate]

I came across the following two definitions of a neighborhood in a topological space $X$. Definition: A set $N\subset X$ is a neighborhood of $x\in X$ if $N$ contains a open set in $X$ which ...
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1answer
37 views

Subspaces that undo Products

I have been working on Munkre's homework sets, and I have come across the following phenomenon: Let $\mathbb{R}_\ell$ be the lower limit topology on the real numbers. If you consider a line as a ...
4
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0answers
41 views

Any two maps to a cone space are homotopic.

I have to prove that any two continuous functions to a cone space are homotopic. Definition of cone space: If $Y$ is any topological space and $I=[0,1]$ is the closed unit interval in $\mathbb R$, ...
4
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1answer
86 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
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2answers
61 views

constructing a CW Complex

I am looking at an example of constructing a CW complex for a space X. The example i am looking at is that for The quotient of $S^2$ obtained by identifying north and south poles. The solution is as ...