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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Sets, Topology and Applying Cantor's Intersection Theorem

I am trying to solve the problem related to the Sierspinski triangle. The triangle is shown as follow. Let $S$ be the intersection of all the finite stages a). Show that $S$ is a nonempty compact ...
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29 views

Cut sets in topology

As part of a problem sheet I have been asked to show that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ whenever $n \neq m$. When I first proved that $\mathbb{R}$ is not homeomorphic to ...
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66 views

Homotopic attaching maps give Homotopy Equivalent spaces

I want to prove that if $f,g : S^{n-1} \to X$ are homotopic maps then the resulting spaces $X \cup_f D^n$ and $X \cup_g D^n$ are homotopy equivalent. I know this question has been asked before: ...
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19 views

Continuity definitions on non-compact subsets

At the top of the wikipedia articles on Hölder condition, Lipschitz Continuity and others, the chain of logic is as follows: On a compact subspace of a metric space: $$Continuously Differentiable ...
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55 views

Interior of sum of sets equals sum of interior of summands

I'd like to have the answer to the following question. If $X_1,X_2\subseteq \mathbb{R}^n$ are convex and compact sets of dimension $n$, does the following hold: ...
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45 views

Proper Maps: Where is continuity used in this Wikipedia proof?

In this article on Wikipedia, a proof is given of the statement that any map $f$ from $X\to Y$ that is closed, continuous, and has the property that $f^{-1}(\{y\})$ is compact in $X$ for $y\in Y$, is ...
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33 views

condition for homeomorphism

If $X $ and $Y $ are homeomorphic as topological spaces is there any necessary and sufficient condition for $X\setminus A$ and $Y \setminus B$ to be homeomorphic?$ A\subseteq X ,B\subseteq Y$
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51 views

$A$ is an interval so $A$ is connected?

I want to prove that if $A\subset \mathbb{R}$ is an interval then $A$ is connected. I found this proof, and I don't understand it essentially the ii) Suppose that $A$ is an interval but not ...
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25 views

How do you show Euler characteristic of any convex polyhedron is $2$?

In the Euler characteristic proof of a convex polyhedron, how do you show the cell decomposition of projection of two polyhedra 1) have a common refinement AND 2) that common refinement comes from ...
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138 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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48 views

Tangent Bundles to manifolds

I am having trouble trying to visualize exactly what a tangent bundle to the klein bottle is spuposed to look like. Is it possible for one to decompose it as a direct sum of simpler bundles?
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54 views

Textbooks to complete concurrently - Self learning empowerment

A user is completing some year challenge that takes them through $9$ textbooks and they are alternating in author. Algebra - Cohn Analysis - Rudin Topology - Lee Repeat three times. I would like ...
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38 views

Separated Spaces and a Partition Differences?

I am just getting a handle on separated definitions from Topology , reading Munkres. So the definition of a separated subsets of a topology, is that they are both disjoint. Further, if each subset ...
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15 views

Closed square homemorphic to the surface of a cube?

Is the closed square in $\mathbb{R}^2$, i.e. $[0,1]^2$ homeomorphic to the surface of the cube in $\mathbb{R}^3$? If they are, is there an explicit homeomorphism? I'm looking for something more solid ...
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20 views

Find all regions formed by a set of circles

I was doodling with Python to draw some circles, and I was able to find all intersection points of a set of random circles, yay ! Now I'm stuck on a question, is there a way to find all regions ...
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15 views

Fundamental domain for a $C_2$-action on a Stone space

The following result seems to be true (I can prove it, only quite indirectly): Let $X$ be a Stone space (i.e. a compact totally disconnected Hausdorff space) and $\sigma : X \to X$ be a ...
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17 views

Topology of the intersection of toric arrangement

Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find ...
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8 views

Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
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46 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
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17 views

closure of rationalsmin topology

Why is the closure of the rationals Q, the reals R ? Is it because the irrational numbers are only considered part of the set of rationals Q in its closure? Aside: Good thorough website to learn ...
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66 views

General definition of piecewise continuity

Is there a general definition of piecewise continuity for functions between topological spaces ? Of course one can intuitively says that $f: X \rightarrow Y$ is piecewise continuous if for every ...
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30 views

Trigonometric polynomials are dense

Is the set of all trigonometric polynomials in the space of continuous functions on [$-\pi,\pi]$ which are $2\pi$-periodic dense?(with sup-norm topology)Please give hints on how to find a sequence of ...
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39 views

Group of Orientation-preserving Homeomorphisms of the Reals.

Let $h: \mathbb{R}\rightarrow\mathbb{R}$ ; $\mathbb{R}$ Reals be an orientation-preserving homeomorphism. I can see $h$ includes linear maps $h=ax+b$ with $a>0$ . Can we say that every ...
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52 views

Face post of a subcomplex complement

Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\simeq$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that ...
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91 views

Definition and Intuition of a Weakly Dense Set

What does it mean to say: set A is "weakly dense" in a set B? The definition of a "dense set" is rather intuitive: the classic example of Q (rationals) being dense in R (reals) is very clear. How ...
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43 views

Conceptual proofs to seven color theorem of torus for 17-19 year olds

what is the best way to explain the seven color theorem of torus to some high school kids and freshman college people?
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39 views

question about Skorokhod distance

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$ ...
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60 views

Is $\mathbb{Q}^{\infty}$ first countable?

Give $\mathbb{R}^{\omega}$ the box topology. Let $\mathbb{Q}^{\infty}$ denote the subspace consisting of sequences of rationals that end in an infinite string of 0's. I'm thinking that since ...
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56 views

Proving that the set is closed.

We use the sequential definition to prove a set is closed. So no continuity or closure or anything related to the topology of the set is allowed. Show $A = \{ x \in \ell^2: |x_n| \leq 1/n \}$ is ...
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18 views

Why $\prod_{n\in \mathbb N}\left([-2^{-n},2^{-n}]\cap\mathbb Q\right)\subset l_2$ is countable and dense in $l_2$?

Why $$\prod_{n\in \mathbb N}\left([-2^{-n},2^{-n}]\cap\mathbb Q\right)\subset l_2$$ is countable and dense in $l_2$? Please explain. My attempt I just try to relate with $\mathbb Q ^\mathbb N$. But ...
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85 views

Finite graph products of finite groups have free subgroup of finite index

This is a problem in Hatcher's Algebraic topology. Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of $K\Gamma$ ...
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31 views

$\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$ dense in $C_0(\mathbb{R}_+)$.

Let $\mathbb{R}_+ := [0,\infty )$ and let $w \in C_0(\mathbb{R}_+)$ be any function with $w(x) \neq 0$ for all $x \geq 0$. Why is $\text{span} \{ e_t \cdot w : t \in \mathbb{R} \}$, where $e_t(x) := ...
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27 views

Explain the Concept of “endedness”

Particularly spaces that are one-ended, two-ended, ... $k$-ended. Can anyone explain via simple examples? Also why two spaces with different ended-ness are not isomorphic.
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39 views

Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
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70 views

Hilbert's hotel with uncountably infinite rooms: can you fit $\mathbb R^2$ guests?

I'm trying to expand on Hilbert's paradox. The original version states that: Suppose there is a hotel with a countable infinity of rooms (eg. $\mathbb N$), all of which are occupied. ...
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alternative to limit of a mapping between topological spaces.

Let $f$ be a mapping between two topological spaces $X$ and $Y$. $\lim_{x \to x_0} f(x) = y$ is defined as for any open set $U_y$ containing $y$, there exists an open set $V_x$ containing $x$, so that ...
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38 views

$f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$

I want to prove that if $f:U \rightarrow \mathbb{C} $ is continuous on $U$ if and only if {$z \in U| f(z)\in V$} is open for every open set V in $\mathbb{C}$. This is my rather incomplete approach to ...
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64 views

Point-set topology - derived sets, boundaries, etc.

Consider the metric space $(\mathbb R^2,\sigma)$, where $\sigma$ is the discrete metric. Let $$E=\lbrace(x,y)\in \mathbb R^2 : x^2+y^2<1 \rbrace$$ i.e. the unit disc. Find the following sets ...
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39 views

Is there a model of set theory in which $2^{2^{\omega_1}}$ is separable?

We know that $2^{\mathfrak c^+}$ ($\mathfrak c =2^\omega=|\mathcal P (\omega)|$) is not separable by the following argument: Suppose $D$ is countable dense in $2^{\mathfrak c^+}$. For each ...
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75 views

Path-connected, simply connected subsets of $\mathbb{R}^n$

A discussion in my topology class caused me to have the following question: Given $A,B \subseteq \mathbb{R}^n$, where $A$ and $B$ are both path-connected and simply connected, need $A$ and $B$ be ...
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47 views

Why are empty measurable spaces and empty topological spaces not desirable?

The definition of a $\sigma$-field $\mathscr{F}$ on a set $X$ (or $\sigma$-ring) requires $\mathscr{F}$ to be a non-empty subset of $\mathscr{P}(X)$. Why is this convention taken? What is the issue ...
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72 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
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178 views

Regarding metrizability of weak/weak* topology and separability of Banach spaces.

Let $X$ be a Banach space, $X^*$ its dual, $\mathcal{B}$ the unit ball of $X$ and $\mathcal{B}^*$ the unit ball of $X^*$. The following result is well-known Theorem. If $X$ is separable, then ...
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59 views

Problem about compact subspace of Hilbert cube.

This is my problem: I have already completed part (i), but I really can't see how I can relate compact subspace with homeomorphism in part (ii). Please give me some ideas.
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Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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64 views

Are Borel sets preserved by an open continuous map?

Does an open, continuous function defined on a compact metric space to itself send Borel sets to Borel sets?
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209 views

The topology generated by open intervals of rational numbers

Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$ Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of ...
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69 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
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26 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
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48 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...