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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23 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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17 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
47 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
26 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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37 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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23 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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34 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
27 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
24 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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26 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
18 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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23 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
62 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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20 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
37 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
42 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
33 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
39 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
34 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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24 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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22 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
64 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
49 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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68 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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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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74 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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0answers
18 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
37 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 ...
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1answer
62 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 ...
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1answer
23 views

Relation of local base with base

Is it true that local base at point $x$ of a topological space is the collection of base elements which contain the point $x$ ?
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1answer
9 views

Is there any metric $d$ of $\mathbb R^n$, $n<\infty$ such that $\mathbb R^n$ is bicompact and no norm induces $d$

There are some simple metrics can't yielded by norm .But add bicompact,I can't structure such example. In fact ,I want to know the condition of metric can be yielded by norm. Sorry for my poor ...
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12 views

Circle rotation number invariant under topological semi-conjugacy.

For a circle homeomorphism $f: S^1 \rightarrow S^1$ we can define the the rotation number $$ \rho(f) = \lim_{n \rightarrow \infty} \frac{1}{n}(F^n(x) - x) \mod 1, $$ for a lift $F:\mathbb{R} ...
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48 views

An odd subset of $\mathbb Q$

The problem is for my topology course, and it is to find a subset of $\mathbb Q$ which is neither open nor closed relative to the real line, but is clopen relative to $\mathbb Q$. I noticed ...
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2answers
16 views

For $B _{n-1 } $ open, $\bar B _n \setminus B _{n-1 } $ is a closed set.

Let $B _n $ be an increasing sequence of open sets in some measure space $X $. Is it true that $\bar B _n \setminus B _{n-1 } $ is a closed set. Can I reason as follows? In the subspace $\bar B _n $ ...
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52 views

Scalar multiplication topology [on hold]

Let $X$ be a topological algebra (i.e. a vector space with a topology where for every $x,y\in X$ then $xy\in X$)and for $k\in F$ and $x\in X$ there are semi open neighborhoods i.e. Let (X, τ) be a ...
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1answer
22 views

Limit points and boundary points of a general metric space

Let X be a metric space, E a subset of X, and x a boundary point of E. It is clear that if x is not in E, it is a limit point of E. Similarly, if x is in E, it is a limit point of X\E. And there ...
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1answer
43 views

Understanding Closure, examples

First thank you in advance. I am reading Munkres book on Topology pg 96 example 6 gives some examples of sets which are closed. I do not fully understand why these examples are closed, and am looking ...
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44 views

Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
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1answer
49 views

how to find a metric to make a space complete (help)

Hi everyone I'm struggle with the following. Define a complete metric on $\mathbb{R}\setminus \{0,1\}$ with usual relative topology. I'd like to follow the big hint of Daniel Fischer but I have ...
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3answers
28 views

Uniqueness of a continuous extension of a continuous map from a set to its closure

Suppose $f$ is a continuous map from a space $A$ to a Hausdorff space Y. Then I know that $f$ can be extended uniquely to a continuous map from closure of $A$ to Y. What is a counterexample to the ...
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1answer
26 views

Extension of a homeomorphism

Let X and Y Hausdorff normal topological spaces, and let N,M dense subspaces of X,Y, respectivaly. Let f a homeomorphism between N and M. Is true that exists an continuos extension F (between X and ...
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72 views

Show that a set $S$ is closed if and only if $S=\operatorname{int}(S)\cup \operatorname{Boundary}(S)$

Show that a set $S$ is closed if and only if$S=\operatorname{int}(S)\cup \operatorname{Boundary}(S)$. I´m using the following definitions: A set is closed if it is the complement of an open set. ...
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31 views

A homeomorphism preserves irreducible components?

Let $f$ a homeomorphism between two Hausdorff topological spaces $X$ and $Y$. Assume that $X$ and $Y$ are reduced analytic spaces. Is true that $f$ takes an irreducible component of $X$ in an ...
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1answer
26 views

Understand the definition of a neighborhood of a point in a topological space

I'm a physics grad student and recently I decided that I would properly learn differential geometry. I then decided that I would start to learn what a topological space is then build up from there ...
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1answer
30 views

Question about space filling curves

I was recently taught that the Peano curve is an example of a continuous bijection from the closed unit interval to the closed unit square. However, if we take a point in the square and take it's ...