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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Proof of topological isomorphism

I remember reading in a section in plato.stanford.edu that the interval $(-∞, t)$ is topologically isomorphic to the interval $(0, t)$. I am not that good with topology, so could someone show me the ...
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1answer
33 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
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1answer
16 views

Finding simple homotopy type

I have an excercise that I kind of dislike: Given $T-\{p,q\}$ where $T = S^1 \times S^1 $ and $p,q \in T$ two different points, I am supposed to find a simple homotopy equivalent space by ...
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0answers
15 views

Transitivity of smooth submanifolds

I was reading through Guillemin and Pollack and was having trouble verifying this for myself. Given $M \subset N$ and $N \subset P$, where $M$ is a submanifold of $N$, and $N$ a submanifold of $P$, ...
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15 views

Define the 4 types of interval subsets of the real numbers.

Define the 4 types of interval subsets of the real numbers. Is the union of an arbitrary number of open intervals also an open interval? Is the intersection of an arbitrary number of open intervals ...
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1answer
36 views

Example of sigma algebra that is not a topology

There is a very nice explanation of an example of sigma algebra that is not a topology: here. I do not fully understand the answer. Apparently this is a basic question, but why do we want this ...
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16 views

Is there a reason why M can't be all summable sequence?

Let M be the set of all summable non-negative sequences $\{x_k\}_{k=1}^\infty$ of real numbers, that is, $x_k \geq 0$ for all k and $\sum_{k=1}^\infty x_k$ converges to a real number. Let $d:M \to ...
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1answer
38 views

Can measure induce a topology on a Set?

When I was taught metric spaces in Topology, I came across the idea that metric defined on a set can induce a topology by creating a basis (open balls). If we have a measure defined on a set, can it ...
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2answers
49 views

What is $d(\sin(x),\cos(x))$ if d is a distance function in a metric space?

Let $M=\{f:[a,b] \to \textbf{R} | f \,is \,continuous \}$. Let $d:M \to \textbf{R}$ be defined by $d(f,g)=\int_a^b |f(x)-g(x)| \,dx$. What is d represent geometrically, and show that M, d is a metric ...
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0answers
39 views

Equivalence of Lebesgue Measurablity

Hello Mathematics Community. I am having some difficulties with the following problem dealing with Lebesgue Measure and its equivalent interpretation. I will first include the definitions which I am ...
2
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2answers
73 views

Why does the product topology allow proper subsets for only finitely many elements?

Consider Theorem 19.1 from Munkres' topology: The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The ...
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44 views

How to organize my learning in Maths?

I m working on a problem in mechanics of material which concerns about the variation of shapes. I need to understand the deformation of material. I m a civil engineering graduate. All my understanding ...
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2answers
74 views

Constructing a circle from a square

I have seen a [picture like this] several times: featuring a "troll proof" that $\pi=4$. Obviously the construction does not yield a circle, starting from a square, but how to rigorously and ...
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1answer
30 views

Two definitions of connectedness: are they equivalent?

A topological space $(X, \tau)$ is connected if $X$ is not the union of two nonempty, open, disjoint sets. A subset $Y \subseteq X$ is connected if it is connected in the subspace topology. In ...
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1answer
40 views

How Can I prove the three statements are equivalent?

Let $X$ be a compact Hausdorff space and $f:X \rightarrow Y$ be a quotient map. Show that the following are equivalent: (a)$Y$ is an Hausdorff space, (b)$f$ is closed map, (c)The set ...
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0answers
23 views

Finding the identification topology

I'm reading the book of Dugundji under Identification topology, and as stated: Let $p: I \rightarrow {\{0\}\cup \{1\}}$ be the characteristic function on $[1/2,1]$. Then the mapping should i think ...
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0answers
35 views

Point-set topology, characterizing sets

I think the question sort of speaks for itself. Please no solutions in the answers - I'm mostly looking to see if my logic makes sense: Let $E$ be a set in a metric space $(X,d)$. Let $E^{\circ}$ ...
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1answer
33 views

Classification of proper maps in topological spaces

How can I prove that if $f:X \to Y$ is continuous of locally compact, Hausdorff topological spaces, then $f$ is proper (inverses of compact sets are compact) iff it extends continuously as a map ...
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2answers
52 views

Determine if $\mathbb{R}$ \ $\mathbb{N}$ is open closed or neither

Determine if $\mathbb{R}$ \ $\mathbb{N}$ is open closed or neither. I've been on this problem for a while now. As of right now I pretty confident its neither because I dont really see how it can be ...
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1answer
28 views

Isolated points are open if $|x| \geq 2$?

Theorem: Define $X$ to be a topological space with $|X| \geq 2. $ Then $x \in X$ is an isolated point$\iff$ $\{ x \}$ is open. I am reading this and the proof proceeds with a neighbourhood $U$ of ...
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1answer
25 views

The closed set in the product topology

We know that for the product topology $X\times Y$, the open sets are generated by $U\times V$,where $U,V$ are open in $X,Y$ respectively. I am considering the closed sets in $X\times Y$, are they ...
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0answers
47 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
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0answers
27 views

Showing a subset of $\;\Bbb R^2\;$ cannot be the set of limit points of any other set

I will appreciate any insight in the following proof (if, indeed, it is a proof): Let $$F:=\left\{\;(x,y)\in\Bbb R^2\;;\;\;xy\in\Bbb Q\;\right\}$$ Prove that there doesn't exist $\;A\subset\Bbb ...
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2answers
37 views

Product metric spaces is again a metric space

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let: $$ d_2 ((x_1,y_1),(x_2,y_2)) = \left[d_X(x_1,x_2)^2 + d_Y (y_1,y_2)^2 \right]^{\frac{1}{2}} $$ for the points $(x_1,y_1)$ and $(x_2,y_2)$ in $X ...
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1answer
29 views

The image of Banach space under its embedding provided by the Banach-Mazur theorem

It is a very nice argument of Banach and Mazur which they use to show that every Banach space $X$ is isometric to a subspace of the space $C(B_{X^*})$, where $B_{X^*}$ is the unit ball of the dual ...
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1answer
26 views

a covering map is open?

$E,B$ are topological spaces and lets say that $p:E\to B$ is a covering map. $p$ is open? i tried to show it as follows: let $U$ be an open set in $E$, and now for every $x\in p(U)$, $p(x)\in B$ ...
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0answers
28 views

Formula for the complement of the Cantor set

According to the wikipedia article: $$C=[0,1] \setminus \bigcup_{m=1}^\infty \bigcup_{k=0}^{3^{m-1}-1} \left(\frac{3k+1}{3^m},\frac{3k+2}{3^m}\right)$$ "Let us note that this description of the Cantor ...
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1answer
23 views

Mapping on induced topology and distance metric

Let $(X, d)$ be a metric space. Let $τ$ be the metric topology on $X$ induced by $d$. For $A ⊆ X$ , let $d(x, A) := \inf_{a∈A} d(x, a) $ for $x ∈ X$ (a) If $f (x) := d(x, A)$ (for a fixed subset ...
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0answers
27 views

Determine the interior, boundary, exterior and closure of the set $S= \{(x_1,…,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$

Determine the interior, boundary, exterior and closure of the set $$S= \{(x_1,...,x_n)\in\mathbb R^n\mid \forall x_i\in \mathbb Q\}$$ I´m using the following definitions: A set is closed if it is ...
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2answers
69 views

Let $S \subset \Bbb R^n$ be non-empty. Prove that $\partial S \neq\varnothing$.

I've been stuck on this one for a while, any help on how to prove this? Let $S \subset \Bbb R^n$ be non-empty. Prove that $\partial S \neq\varnothing$.
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0answers
21 views

How to draw Congressional districts to mirror the Popular Vote

Let me preface this by saying that I'm not sure whether this is fundamentally a mathematical question or not, but I think it is. In the United States, the House of Representatives is elected roughly ...
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1answer
38 views

A topological space is extremally disconnected iff every two disjoint open sets have disjoint closures

Show that for any topological space $X$ the following are equivalent: $X$ is extremally disconnected Every two disjoint open sets in $X$ have disjoint closures. My attempt at a ...
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2answers
43 views

Fundamental group of two tori with a circle ($S^1✕${$x_0$}) identified

Compute the fundamental group of the space obtained from two tori $S^1✕S^1$ by identifying a circle $S^1✕${$x_0$} in one torus with the corresponding circle $S^1✕${$x_0$} in the other. Using van ...
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1answer
34 views

Empty union existence? A basis for the topology

Consider $(X, \tau)$ where $$\tau = \{\emptyset, X, \{a \}, \{c\}, \{a,c \},\{a,b \},\{b,c \} \},$$ and the nonbasis $$B = \{ \{a\}, \{c\},\{a,b\},\{ b,c\} \}.$$ My book says $\emptyset$ can be ...
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1answer
43 views

Relative compactness of metric space

I know that in a metric space $X$ compactness, countable compactness and sequential compactness of a subspace $X'$ are equivalent using the definition of countable compactness as every infinite subset ...
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1answer
36 views

Construction of a covering space as a fibre bundle

In a direct proof of the equivalence of categories between the covering maps $p:(\hat X, \hat x) \rightarrow (X,x)$ of a topological space $(X,x)$ for sufficiently beautiful $X$ and the ...
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2answers
34 views

Wedge sum of spheres [on hold]

Let's $X$ be a CW-complex. If $X^{(n)}$ is the n-skeleton of $X$ and $\Lambda_n$ is a set of index. How could I prove that $X^{(n)}/X^{(n-1)}=\bigvee_{\alpha \in \Lambda_n} S^n_{\alpha}$? Thank you ...
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2answers
26 views

Topology vs Borel sigma-algebra on a set X

What is the difference between: (X: a set) Topology (open set system) on X Borel sigma-algebra on X Both are a set of open subsets. Both include X and empty set. Both are Closed under union and ...
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1answer
24 views

Relative compactness implies relative countable compactness?

By using the fact that compactness implies countable compactness, I think that relative compactness implies relative countable compactness in any topological space. Am I right? Thank you so much!
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0answers
24 views

Let (X, T) be a topological space. If U is in T, do we use the notation for U is an element of T or can we also say U is a subset of T?

I know this is a basic question, and I am pretty certain that T, the collection of all open sets has only elements rather than subsets (at least not in this context). Could someone clarify? I know the ...
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1answer
65 views

Is $d(x,y) = (x-y)^2$ a metric on $\Bbb R$?

For $x,y,z \in \Bbb R$, define $d(x,y):= (x-y)^2$ Is this a metric on $\Bbb R$? It's clear that $d(x,x)=0$ and $d(x,y)=d(y,x)$ for all $x,y \in \Bbb R$. The triangle inequality seems to have a ...
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2answers
51 views

Suppose $x$ is a limit point of $A \subset X$, then if $f: A \to Y$ is continuous, is it true that $f(x)$ is a limit point of $f(A)$?

So I already know that a counterexample is $f(x) = c$ for $c$ is a constant, but I can't seem to prove this statement by contradiction, all I did was go back and forth. "Proof": If $f(x)$ ...
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1answer
69 views

Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
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2answers
48 views

How to Prove: If a function $f$ is continuous on a compact set $K$, then $f$ is bounded on $K$ [on hold]

How to Prove: If a function $f$ is continuous on a compact set $K$, then $f$ is bounded on $K$ ($f(K)$ is a bounded set) plz let me know.
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2answers
76 views

$\mathbb{Q}$ can not be embedded in $\mathbb{Z}$

Show that $\mathbb{Q}$ can not be embedded in $\mathbb{Z}$ (where both has the subspace topology of $\mathbb{R}$) My attempt at a solution Since Z is discrete, {k} is open in $\mathbb{Z}$ with ...
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1answer
23 views

Mobius band parameterizaton: Showing injective

So, I'm trying to show that the parameteization function from $\mathbb R^2$ to $\mathbb R^3$ given in the wikipedia page http://en.wikipedia.org/wiki/Mobius_band#Geometry_and_topology is injective on ...
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1answer
51 views

continuity extension of exponential $f(x)= a^x$

Consider tha exponential function $f(x) = a^x$, where $f: \mathbb{Q} \to \mathbb{R}$. My problem is to show that it has unique extension and how am I going to define this one? Also, I used a ...
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2answers
41 views

Obtaining Wirtinger presentation using van Kampen theorem

Hatcher's Algebraic Topology, section 1.2, problem #22 describes an algorithm for computing the Wirtinger presentation of the complement of a smooth or piecewise linear knot K in $\mathbb{R}^3$: ...
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1answer
35 views

Product topology of $\Re^X$ et similia

I have a problem with visualizing how exactly the product topology of something like $\Re^X$ looks like. Just a quick summary of my line of reasoning. Given a family of topological spaces ...
2
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1answer
64 views

A question about locally constant functions.

A mapping $f:X\to Y$ is defined to be locally constant if $\forall x\in X$, there exists a neighbourhood $V(x)$ containing $x$ such that $a\in V(x)\implies f(a)=x_0$ for some constant $x_0$. In other ...