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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39 views

Showing R is Sequential

I want to prove rigorously that R is sequential. Sequential means that every sequentially open set is open (see: http://en.wikipedia.org/wiki/Sequential_space). I can understand intuitively why R is ...
2
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1answer
185 views

Prove that $\mathbb R/\mathbb Z$ is sequential and not first countable

Let's say I have a a space X, the quotient space of R (the reals) obtained by identifying all points of Z (the integers). How do I prove that X is sequential but not first countable? (sequential ...
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2answers
58 views

Proof with compact sets in Hausdorff space

Prove that every compact set in Hausdorff space is closed. Let $(X,\tau)$ be Hausdorff space and $A,B$ compact, disjont subsets of $(X,\tau)$. Prove that exist two disjoint sets $V,W$ open in ...
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0answers
21 views

very elementary question about bases on the real line

Let $\mathcal{U} = \{ (-\infty,a) : a \in \mathbb{R} \} $. I want to show $\mathcal{U}$ is a basis for a topology on the real line. Attempt Let $x \in \mathbb{R}$. Choose $I = (-\infty, x+1)$. Then ...
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1answer
72 views

How do I show R is regular?

How do I prove that R (the reals) are a regular space? This means I need to show that given any nonempty closed set F and any point x that does not belong to F, there exists a neighborhood U of x and ...
2
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2answers
103 views

What does it mean for a set to be relatively open?

The definition is that if we are given that Y is a subset of X (where X is a metric space), a subset E of Y is open relative to Y if and only if E=(Y intersect G) for some open subset G of X. I'm ...
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2answers
37 views

Let X, Y be topological spaces and let y ∈ Y . Show that the map i : X → X × Y, i(x) = (x, y) is continuous

I have a feeling the solution is to do with the pre image of an open set in XxY being open in X, but I'm not sure how to go about proving it.
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1answer
40 views

How to prove that $\mathbb{R}$ is $T_1$?

I want to show directly that $\mathbb{R}$ is $T_1$, meaning that for any two points $x,y \in X$ there exists two open sets $U$ and $V$ such that $x \in U$ and $y \notin U$, and $y \in V$ and $x ...
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2answers
80 views

Showing that $\mathbb{C}$\ {$x+iy|x,y\in \mathbb{Q}$} is connected

This I find it really hard to solve. I suppose the set {$x+iy|x,y\in \mathbb{Q}$} is neither closed or open. But I just cannot seem to find a way to go forward. Can someone help me out. Thanks
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2answers
42 views

Compactness of a sum of spheres in $\mathbb{R}^n$

Let $S(a,t)=\{x\in\mathbb{R}^n : d_e(a,x)=t\}$ be $n$-dimensional sphere in $(\mathbb{R}^n,\cal{T}_e)$ (natural, euclidean topology). Then for $A\subset\mathbb{R}^n$ and $r:A\longrightarrow ...
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1answer
75 views

Topological form of Martin's Axiom

I'm currently studying consequences of Martin's Axiom: Martin's Axiom (MA): Suppose that $\left\langle P, \leq \right\rangle$ is a ccc partially ordered set and $\{D_\alpha\}_{\alpha < \lambda}$ ...
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1answer
46 views

Set of orthogonal matrix over $\Bbb{R}$: Closed, convex, open?

Reading my course on topology we haven't answering this exercise: Show that the set of orthogonal matrix over $\Bbb{R}$ is closed. Is it convex? Open For the fact is closed I wrote ...
2
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2answers
108 views

Basic question on the topologies associated with the real lines

Let $X = \mathbb{R}$. The standard topology is $\mathcal{T}$ in where the open sets are just elements of the form $(a,b)$. Let $\mathcal{T}_L$ be the topology where the open sets are $[a,b)$. Next, ...
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1answer
17 views

showing that a map is a quotient map under certain conditions

Let f : L -> K be a surjective continuous map. If both L, K are compact and Hausdorff, could anyone tell me how to prove that f is a quotient map?
2
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1answer
139 views

Baby Rudin Problem 2.29

Here's is Prob. 29 in the Exercises following Chap. 2 in PRINCIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin, 3rd edition: Prove that every open set in $\mathbb{R}^1$ is the union of an at most ...
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3answers
81 views

show: $\overline{\overline X} = \overline X$

is my proof correct? Definition: Let $X\subset\mathbb R$ and let $x'\in\mathbb R$, we say that $x'$ is an adherent point of $X$ iff $\forall\epsilon>0\exists x\in X \text{ s.t. }d(x′,x)≤ε$. the ...
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1answer
21 views

an injective map can not take several intersecting arcs onto line segment

I read a result in the theory of harmonic mappings, and i think it might be true in general setting as well. But i am unable to get a proof of this. Can anyone help me with proving it. The statement ...
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0answers
32 views

Checking for a function to be homeomorphism

Let $h=(h_1,h_2):U\to \mathbb{R}^2$ be one-one and continuous in a neighborhood $U$ of the origin in $\mathbb{R}^2$ with $h(0)=0$ and $h_1$ harmonic. Then $h:U\to h(U)$ is bijective and continuous. If ...
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2answers
40 views

Distance from a proper subspace of a metric space.

Let $(X,d)$ be a metric space. And $A \subset X$ be a proper closed subspace. Proper here implies every closed and bounded ball is compact. Define $d_A(x)= inf_{a \in A} d(x,a)$. Show that this ...
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1answer
170 views

On the largest and smallest topology on a given set.

Let $\{ \mathscr{T}_{\alpha} \}_{\alpha \in \Sigma }$ be a family of topologies on a given set $X$. Question: How can I find the unique smallest topology on $X$ containing all the collections ...
2
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1answer
116 views

Topology: Convergent subsequence implies compactness

I've looked at the related questions/answers to my problems, but I need clarification. I don't understand the contradiction from the following proof. : I want to show that in the metric space ...
3
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1answer
247 views

Rank Theorem proof

Let $\phi: M \to N$ be an immersion from smooth manifold $M^m$ into $N^n$ ($\dim M = m$ and $\dim N = n$). Prove there exists smooth charts $(U,h)$ in $M$ with $p \in U$, $h(p) = 0$, and $(V,g)$ in ...
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2answers
84 views

Show that the intersection of an open set and the closure of any subset is a subset of the closure of the intersection of both subsets.

Let $A$ be an open set and $B$ be a subset of a topological space $X$. Show that $A \cap cl(B) \subseteq cl(A \cap B)$. I'm trying to show that for a point $x \in A \cap cl(B)$ that $x$ is in all ...
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1answer
153 views

Proving that for a set $U\subset C$, $U$ is open if and only if for all $x \in U$, there exists a region $R$ such that $x \in R \subset U$.

Prove that for a set $U\subset C$, $U$ is open if and only if for all $x \in U$, there exists a region $R$ such that $x \in R \subset U$. I just have no idea how to start and was wondering if ...
4
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1answer
116 views

$d(x_n,y_n)$ converges to a limit when $x_n, y_n$ are Cauchy sequences

Let $(X,d)$ be a metric space and $x_n, y_n$ Cauchy sequences. Is there a way to prove that $\lim\limits_{n \to \infty} d(x_n,y_n)$ exists without involving the completion of $X$? Intuitively you ...
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1answer
78 views

Question related to the definition of accumulation point and Bolzano Weierstrass theorem

I think I dont understand what a limit point of a set is. For example, there are two ways in which I have seen the BW theorem stated: 1. if the set $X$ is compact, then any sequence of points in X has ...
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2answers
77 views

Example of a connected set

Can someone please give an example of a connected set? The formal definition is that if the set $X$ cannot be written as the union of two disjoint sets, $A$ and $B$, both open in $X$, then $X$ is ...
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0answers
106 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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0answers
106 views

Dictionary order topology and subspace topology

Compare $(0,1) \times (0,1)$ with the dictionary order topology to the same set with the subspace topology given by the dictionary order on $\mathbb{R} \times \mathbb{R}$. This is an exercise in my ...
0
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1answer
68 views

Not Quite Metrization

Let's say I have a space $X$ with a function $d\colon X \times X \to \mathbb R$ that has the following 2 properties: $d(x,y)\ge 0$ for all $x$, $y \in X$ and $d(x,x) = 0$ for all $x$, $y \in X$. ...
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2answers
79 views

Projections and open maps

I have the projection $\pi_x : X \times Y \to X$ where $\pi_x$ is defined as: $\pi_x(x,y) = x$. How can I show this is an open map? I know that a map $f: X \to Y$ is an open map if whenever $U$ is ...
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2answers
50 views

Inclusion of smooth maps implies smooth again

Let $\phi: N \to M$ be a continuous map from smooth manifold $N$ to smooth submanifold $M \subset \Bbb R^p$. Let $i :M \to \Bbb R^p$. Show that if $i \circ \phi$ is smooth, then so is $\phi$. ...
2
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1answer
117 views

Proof that an Infinite Simplicial Complex can only have countably many Simplices?

We define an infinite simplicial complex $K$ to be a set of simplices in some $\mathbb R^n$ sch that the following conditions hold: $1$. Given a simplex in $K$, every "face" of A (i.e., the simplex ...
0
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1answer
63 views

How to show that a closed ball doesn't allow stochastical ordering

Given a closed ball $$\cal{F}=\{g:D(g,f)\leq\epsilon\}$$ where $f$ and $g$ are some density functions and $D$ some distance say relative entropy: $$D(g,f)=\int ...
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2answers
47 views

Question regarding the finite intersection property

If I have a sequence $\{C _j \} $ of closed and bounded intervals such that $ \cap _{j=1 } ^{\infty } C _j= \emptyset $ Does it follow that for some $n $, we must have $\cap _{j } ^n E _j = ...
0
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1answer
46 views

Retraction on Connected Space

Does there exist a retraction from a connected topological space to a subspacce with exactly two points. Will somebody please give me a hint for this one or at least a good way to start. Thanks.
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2answers
86 views

Show a bound sequence with a cluster point is indeed convergent

I have been wondering for a while now: How do I show that $(i) \space (a_{n})_{n \in \mathbb{N}}$ is convergent. $(ii)\space (a_{n})_{n \in \mathbb{N}}$ is bounded and has a cluster point. ...
3
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2answers
322 views

How to prove that the quotient space of the punctured plane under dilation is homeomorphic to a torus?

Suppose we're considering the map $f:\mathbb R^2\backslash (0,0)\rightarrow \mathbb R^2\backslash (0,0)$ given by $f(x_1,x_2)=(cx_1,cx_2)$ with $c$ a positive real number. How does one show that the ...
2
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1answer
54 views

Elementary topology question about bases and topologies

Let $X$ be a topological space. Let $\mathscr{O}$ be the collection of open sets of $X$ with the following property: For each open $O \subset X$ and each $x \in O$, there exists and element $U ...
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0answers
83 views

Connected space such that (almost) no subspace is connected [duplicate]

Is there a connected space $(X,\tau)$ such that $X$ has more than $2$ points and the only proper connected subsets of $X$ are the singletons?
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2answers
215 views

Give an example of a set that is closed but not compact nor bounded. Prove your answer.

Let $X = (0,\infty)$ with the usual topology in $\mathbb{R}$ and the the usual metric. Consider $A \subset X$ where $A = [1, \infty)$. Then $A$ is closed as $A' = (0,1) \subset X$. My attempt is as ...
2
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1answer
95 views

Is $\mathbb{A}^1$ isomorphic to one of its quasi-affine subsets?

It's well known that $\mathbb{A}^1$ is not isomorphic to any proper open subset of itself. Just out of curiosity, is $\mathbb{A}^1$ isomorphic to any proper quasi-affine subset of itself, or is this ...
4
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1answer
373 views

Every closed set in a separable metric space is the union of a perfect set and a set which is at most countable [duplicate]

Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin's Principles of Mathematical Analysis, 3rd ...
0
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1answer
72 views

Locally compact Hausdorff space

How can you prove the following statement: If a Hausdorff space $S$ is locally compact, then for any point $a \in S$ and any open neighbourhood $U$ of $a$ there is an open set $V$ in $S$ such that $a ...
3
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1answer
45 views

Computing $H_i(\mathbb{RP}^n \times \mathbb{RP^m}; G)$

I'm trying to compute $H_i(\mathbb{RP}^n \times \mathbb{RP}^m; G)$ for $G = \mathbb{Z}, \mathbb{Z_2}$ respectively by using the cellular chain complexes. I'm not really sure how to get started, ...
0
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1answer
26 views

Is dimension unique?

To what extent can we describe a space as being uniquely n-dimensional? For example, the space of functions on I=[0,1] are frequently described as infinite dimensional, where I serves as the index for ...
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3answers
41 views

Show that there exists $\epsilon >0$ such that $\bigcup_{x\in A}B(x;\epsilon)\subset V.$

Let $X$ be a compact metric space, $A$ a closed subset of $X$ and $V$ an open subset of $X$. Suppose $A\subset V$. Show that there exists $\epsilon >0$ such that $$\bigcup_{x\in ...
1
vote
1answer
128 views

Let X be a subspace of $\mathbb R^2$ consisting of points that at least one is rational. Prove that X is path-connected. [duplicate]

Let X be a subspace of $\mathbb R^2$ consisting of points such that at least one of coordinates x and y is rational. Prove that X is path-connected. A sketch is as follows. Is it right? Also How to ...
1
vote
1answer
37 views

Connectedness of sets

Let $E \subset S$. Suppose $E$ is not connected. Then in the induced topology of $E$ relative to $S$, $E$ and $\emptyset$ are not the only clopen sets. I can show using the above definition that if ...
0
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1answer
93 views

$S^1 \times S^1$ is homeomorphic to torus

The question is asking to show $S^1 \times S^1$ is homeomorphic to a torus. I have read some other posts in here but most of them are proving it with "lattice" which I haven't learn. Here is what I ...