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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In a metric space, prove there is an invertible function $\Bbb R^n\to\Bbb R^n$ such that $f(a)=b$

I would like to prove the following theorem from Mendelson's Introduction to Topology: For each $a,b\in\Bbb R^n$, prove that there is a topological equivalence between $(\Bbb R^{n},d)$ and ...
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32 views

If $S = \{x \in [0,1] \mid f(x) \neq0\}$ and $f$ is continuous on $[0,1]$, why is the complement of $S$ closed?

If the set $S = \{x \in [0,1] \mid f(x) \neq0\}$ and $f$ is continuous on $[0,1]$, why is the complement of $S$ closed? I am unable to see why, is this due to finiteness? Thanks!
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35 views

question about Stone Čech compactification

$x$ is normal space and we recognize him by his picture in $βX$. show that every $c_1 c_2$, close and disjoint sets in $x$ also the closure of $c_1$ and $c_2$ (in the closure of $x$) is disjoint. i ...
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22 views

Proving that the $C_b(M)$ is a complete space with the $L^{\infty}$ norm.

Suppose $A$ is some metric space, and let us define $C_b(M)$ as the vector space consisting of the set of all bounded continuous $\mathbb{R}$ valued functions on $A$. Now, we define the $L^{\infty}$ ...
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88 views

Reflexive normed spaces are Banach

I want to prove that a reflexive normed space $X$ is a Banach space. By the definition of the reflexive space, the evaluation map $J:X\to X''$ is a bijection. All I need is to prove that the ...
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56 views

Questions on connectedness

I have a final examination in general topology this week, and I've been doing past papers for the past two days in anticipation for it. I'm not sure if my answers are correct so could someone tell me ...
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18 views

Finding lifted paths, homotopy lifting

I am given a covering map $p: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R}^2 \setminus \{0,0\}$ defined by $p(r, \theta)=(r \cos 2 \theta,r \sin 2 \theta)$ Let $\alpha: [0,1] \to \mathbb{R}^2 ...
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22 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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21 views

The smallest topology on $X$ containing $\rho_1 \cup \rho_2$ is a compact topology

Let $\rho_1$ be a cofinite topology and $\rho_2$ be any compact topology on an infinite set $X$. Show that the smallest topology on $X$ containing $\rho_1 \cup \rho_2$ is a compact topology.
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20 views

Prove that $h$ and $p*q$ are homotopic relative to {$0,1$}

Let $0<s<1$. Given paths $p$ and $q$ with $p(1)=q(0)$, define $h$ by the formula $$h(t) = \begin{cases} p(t/s),& \text{if} \quad 0 \leq t \leq s \\ q((t-s)/(1-s)), &\text{if} \quad ...
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29 views

A space $X$ is path connected if and only if there is a point $a$ in $X$ such that each point of $X$ can be joined to $a$ by a path in $X$.

A space $X$ is path connected if and only if there is a point $a$ in $X$ such that each point of $X$ can be joined to $a$ by a path in $X$. I'm trying to prove this statement. The only if direction ...
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17 views

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property.

Prove that intervals of the form $(a,b]$, $[a,b)$, $(-\infty,a]$, $[a,\infty)$ do not have the fixed point property. In the case of open intervals, I can derive that they do not have the fixed point ...
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24 views

G is open in X iff $\overline{G \cap \bar{A}}=\overline{G \cap A}$ for all $A\subset X$

It is clear that $\overline{G \cap {A}}\subset \overline{G \cap \bar{A}}$ but not getting how to show the reverse inclusion!
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48 views

Is Alexandroff Duplicate A(X) of X paracompact?

Prove or disprove: If $X$ is a paracompact space, then Alexandroff Duplicate $A(X)$ of $X$ is paracompact. Thanks for any help. ...
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22 views

Approximation of a Function Discontinuous only at a Set of First Category

Consider a function $f$:$T$ $\rightarrow$ $R$ that is continuous except for a set of first category. Brosowski and da Silva (1997) argue that there exist a sequence of continuous functions that ...
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33 views

Examples of continuous integer-valued functions on totally disconnected spaces

I wanted to see examples of continuous integer-valued functions $f:X\to \mathbb{R}$ on a totally disconnected space $X.$ I have only some abstract examples in mind.
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29 views

Covering a cube with open balls centered at lattice points

I'm trying to prove that given $\epsilon >0$, the balls $B(\epsilon j;\epsilon)$ cover a cube of the form $T = [-b,b]^n$, where $j=(j_1,...,j_n)$ ranges over all integral lattice points of $R^n$ ...
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34 views

Compact and connected

Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by $$\{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...
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24 views

A intersection interior of B is a subset of interior of B relative to A

I was able to prove that $A \cap int(B) \subseteq int_A (B)$ but I wasn't able to show the opposite direction. Please help me. If $int_A(B)\nsubseteq A\cap int(B)$, what condition is required for it ...
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20 views

Showing that a k-cell is totally bounded.

I want to prove that a k-cell or cube of the form $T$ = $[-b,b]$ x $\cdots$ x $[-b,b]$ for some $b > 0$, is totally bounded. The reference that I am looking at suggests the balls $B(\epsilon ...
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61 views

Interior and closure of an arbitrary product

Let $J$ be a non-empty subset of the set $\mathbb{N}=\{1,2,3,...\}$ of natural numbers and consider $X = \mathbb{R}^J$ with the product topology. $(1)$ Show that the following statement holds: ...
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26 views

Necessary and sufficient condition for existence of a deck transformation

I am considering the following problem: $\tilde X$ path connected, $X$ path connected, locally path connected, $P:\tilde X \to X$ covering map, $x_0 \in X, \tilde x_0, \tilde x_1 \in p^{-1}(x_0).$ I ...
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69 views

Liftings in Covering Map Closed

Let $p:\tilde{X} \mapsto X$ be a covering map with $\tilde{X}$ path connected. Why are all liftings of a closed path $f$ in $X$ either closed or not closed? If $\omega$ is a path from $\tilde{x_0}$ ...
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82 views

Is there a continuous function from X onto Y?

(1) $X= (0,1]$ and $Y = [0,1]$; (2) $X = [0,1]$ and $Y$ is the topologist's sine curve (3) $X = [0,1] \cup [2,3]$ and $Y = X \times X$ I believe there is a continuous function for the first one. I ...
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35 views

find the interior, closure, and boundary

$\ A =\{(\tfrac{1}{m},\tfrac{1}{n})\in \mathbb{R}^2 : m,n \in \mathbb{Z}\backslash \{0\}\}$ $\ C =\cup\space B((\tfrac{1}{n},n),\tfrac{1}{n}) \space for\space n\in \mathbb{N}$ My claim is $\ ...
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15 views

Show that any path $Y, \tau'$ in a $T_2$ space is second countable.

We may use the fact that any continuous function from a compact $T_2$ space onto a $T_2$ space is closed. A hint I received is let $\{U_n\}, n \in \mathbb{N}$, be a countable basis for the usual ...
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22 views

I need help with understanding this proof that the product topology on $\mathbb{R}^n$ is the same as that induced by the square metric

Here's the proof: I'm basically struggling with following the logic used in the proof. How is for each $i$ there an $\epsilon_i$ such that $(x_i - \epsilon_i, x_i + \epsilon_i)$ is contained in ...
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17 views

$L \cap V = \overline{L} \cap V$ when $L\cap V$ is closed in $V$.

Let $E$ be a topological space and $L, V \subset E$, $V$ open, and $L \cap V$ closed in $V$, then $\overline{L} \cap V = L \cap V$. Attempt: $L \cap V$ closed in $V$ implies $L \cap V = F \cap V$ for ...
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53 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
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81 views

Does maximal principle imply open mapping theorem for any continuous function?

At first I spent a lot of time looking for counterexamples because I had never seen such a claim that M.P. implies O.M.T.. But later I realized the claim might be true, so I just had a try and proved ...
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45 views

What does 'real-valued' function mean in topology?

If I have a topological space $X$ and a 'real-valued' function $f$ on $X$. Does this mean I have a map of the form: $f: X \rightarrow \mathbb R$ where $\mathbb R$ has the usual topology? Or something ...
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28 views

Showing that a map can be deformed into the identity.

Suppose $F((a,b), k) = (ae^{\pi i k}, be^{\pi i k})$ where $0 \leq k \leq 1$. Now would $g(a,b)$ = $(-a,-b)$ if $g : S^{1} \rightarrow S^{1}$?
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47 views

Show that a subspace of $S \times S$ is not separable

I'm trying to show that on $S \times S$ where $S$ is the Sorgenfrey line, the subspace $Y=\{(-x,x)\}$ is discrete so I can show that $Y$ isn't separable. I'm taking $u=[-x,a) \times [x,b)$ such that ...
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22 views

maximal independent set in a graph

Let $G$ be a graph and $A$ is a subset of vertex set of $G$. $A$ is said to be independent if for any $x, y \in A$, $(x,y) \notin E(G)$, i.e $x$ and $y$ not connected by an edge. Further A is said to ...
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48 views

Why is this map closed?

I've encountered with this question during reading J.M.Lee's book: Define an equivalence relation on $\mathbb{R}^{2}$: $(x,y)\sim(x',y')$ iff $(x',y')=(x+n,(-1)^{n}y) $ for some $n\in\mathbb{Z}$ ...
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45 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
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60 views

Fundamental Group equaling 0

Let $X$ be a space for which $\pi(X,x)=0$. If $f,g$ are two paths in $X$ with $f(0)=g(0)=x$ and $f(1)=g(1)$, why is $f$ equivalent to $g$?
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60 views

Show homeomorphism between convex hull and unit ball?

In the proof of Schauder fixed point theorem in Evans' PDE book, he uses a claim that the convex hull $K$ of $N$ points $x_1,\dots,x_N$ in a convex compact subset $A$ of a Banach space $X$ is ...
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18 views

continuity of product path--pasting lema

A path in a topological space $X$ is a continuous function $f:[0,1]\longrightarrow X$. For two paths $f,g$ in $X$ such that $f(1)=g(0)$, the product path $f*g:[0,1]\longrightarrow X$, is defined by ...
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65 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
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46 views

proposition about boundary points of subset of topological space

Let $(X,\mathcal T)$ be a topological space and let $A$ be a subset of $X$. Then: $A$ is closed if and only if $\partial A$ is a subset of $A$ $A$ is open if and only if $A\cap\partial A=\emptyset$ ...
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30 views

How is $\displaystyle{x\times (0,1)}$ open in $I_0^2$?

Take $I_0^2=[0,1]\times[0,1]$. How is $\displaystyle{x\times (0,1)}$ an open set in $I_0^2$? Thanks.
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40 views

given a basic neighborhood space, finding an algorithm that will determine the open sets of that space.

I would like to figure out how to find open sets of the basic neighborhood space ($\mathcal{B}_X,X$) where $X$ is the space and $\mathcal{B}_X$ is the basic neighborhood on $X$. The second to last ...
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66 views

Confusing Notations in a Paper

From : "Multicast Routing and Design of Sparse Connectors" by Andreas Baltz and Anand Srivastav, Springer 2009: http:\\link.springer.com/chapter/10.1007%2F978-3-642-02094-0_12 In 2-Copy Method, ...
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36 views

How can I prove formally that the projective space is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: Define $\mathbb{P}^n$, the real projective space of dimension n to be the set of 1-dimensional linear subspaces ...
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43 views

Cartesian product of open sets

Say I am dealing with the standard topology $\tau$ on $\mathbb{R^2}$ and I have the sets $a , b \subset \mathbb{R}$ where $a = (2, 4)$ and $b = (5, 6)$. Then does it mean that $a \times b \subset ...
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25 views

A stronger concept than total boundness

A space, every proper principal filter of which is refined by a Cauchy filter, is called totally bounded. Is there a term (and theory) about a stronger concept: a space every proper filter of which ...
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35 views

Interior Points Confusion

I am confused about interior points. Basically, I think that if the set we are not working in doesn't contain irrational numbers, then the interior of the set is $\emptyset$. $\left [ 0, \; 5\right ...
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1answer
42 views

Closedness of the closed half-space

Suppose we have a hyperplane H(p, α) = {x ∈ R n | p · x = α} , then how do we prove that one of the corresponding closed half-spaces, H*(p, α) = {x ∈ R n | p · x ≤ α} is indeed closed? For every x ...
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33 views

triangle inequality on a given metric

$X$ be set consisting of all sequences $(x_1,x_2, \dots)$ s.t $x_i \in \mathbb R$ and $\sum x_i^2$ converges I need to prove triangle inequality for the metric on $X$ given by, $d(x,y) = [ ...