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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Why do we need surjectivity in this theorem?

In class we proved the following theorem: Given $X_1,X_2$ ordered sets. Then any surjective increasing $\phi: X_1 \to X_2$ is continuous wrt the interval topology on $X_1$ and $X_2$. I was asked to ...
3
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1answer
78 views

Approximation of a strongly measurable function by a sequence of simple functions.

Let $(X, \mathcal{A})$ be a measurable space and let $E$ be a normed space. $(i)$ $f:X \rightarrow E$ is called Borel measurable if $f^{-1}(B) \in \mathcal A$ for all $B \in \mathcal{Bo}(E)$ where ...
3
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1answer
107 views

Are $L^\infty$ bounded functions closed in $L^2$?

Is the set $\{ m \in L^2(0,1) : |m|_{L^\infty}\leq A \}$, (i.e. the set of $L^2$ functions with bounded $L^\infty$ norm) a closed subset of $L^2$? (Closed in the topology induced by the $L^2$-norm)
3
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1answer
88 views

Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.

Question: Let $(X, d)$ be a metric space. Let $A$ and $B$ be disjoint subsets of $X$ that are closed in the topology induced by $d$. Prove that there exist disjoint open sets $U$ and $V$ such that ...
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1answer
74 views

“isometric invariant” vs “isometric” what do these term mean?

I am now hopelessly confused: There is Hilberts Theorem https://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29 . that implies that there are no isometric embeddings of the ...
3
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1answer
95 views

Build a topological manifold starting from a set.

Suppose you are given a generic set $X$. There exist sufficient and non-trivial conditions that ensure the existence of a topology $\tau_X$ on X such that the topological space $(X,\tau_X)$ is a ...
3
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1answer
124 views

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$.

Proof that the cardinality of continuous functions on $\mathbb{R}$ is equal to the cardinality of $\mathbb{R}$. I think is should be proved with the help of Cantor-Bernstein theorem. It is easy to ...
3
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2answers
153 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
3
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2answers
119 views

Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
3
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2answers
100 views

Connected topological spaces, product is connected

Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is ...
3
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2answers
276 views

If two Borel measures coincide on all open sets, are they equal?

Let $X$ be a topological space and let $\mathcal{B}(X)$ be its Borel $\sigma$-algebra. That is, $\mathcal{B}(X)$ is the smallest $\sigma$-algebra on $X$ containing all the open sets. Let $\mu, \eta : ...
3
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1answer
119 views

Independence of $H(f)=\int_M \alpha \wedge f^* \beta$ on choice of $d\alpha=f^*\beta$?

I came across the following UCLA qual question while studying for my upcoming qual: Let $f: M^{4n-1} \to N^{2n}$ be a smooth map between closed connected oriented manifolds of the indicated ...
3
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2answers
82 views

If $(X,\| \|)$ be a Normed Linear Space.. Show that if any vector subspace $Y$ of $X$ is open, then $Y=X$

If $(X,\| \|)$ be a Normed Linear Space, Show that if any vector subspace $Y$ of $X$ is open, then $Y=X$ Attempt: Subspace $Y$ of $X$ is open, $=> \exists ~~r >0 ~~\forall~~ y \in Y$ s.t $ ...
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2answers
88 views

Algebraic Object in Topology

Common algebraic categories are group, ring, module and algebra. Some of them have the corresponding object in topology, like topological group and topological linear space. We define them by making ...
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1answer
293 views

Universal Cover of a Surface (with Boundary)

I'm trying to see if there is a "nice-enough" way of describing/constructing the universal cover for a compact surface with n boundary components. Clearly, if $n=0$ , the classification theorem for ...
3
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1answer
801 views

Homeomorphism between punctured plane and cylinder [duplicate]

I am asked to prove that the cylinder and the punctured plane are homeomorphic. I understand that I need to find a function that maps every point in the plane to a point on the cylinder. I can ...
3
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2answers
387 views

Locally compact subspace is an intersection of an open and closed set

Let X be a locally compact topological space. I need to prove that if $M\subset X$ is a locally compact subspace of X then there exist $U,F\subset X$ such that U is open and F is closed, and $M=U\cap ...
3
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1answer
151 views

“Proof” that $g(t) = (t,t,t,…)$ is not continuous with uniform topology

Let $g : \mathbb R \to \mathbb R^{\omega}$ be the function $$ g(t) := (t, t, t, \ldots). $$ If $\mathbb R^{\omega}$ is equipped with the uniform topology, and $\mathbb R$ with the standard topology, ...
3
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1answer
173 views

Is a simply connected set connected?

A set $X$ is considered connected if there is no separation of the set $X$ into disjoint sets $A,B$ such that $X = A \cup B$, where neither sets ($A$,$B$) contain limit points of each other. Now a ...
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0answers
164 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
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2answers
334 views

Trying to understand a “Scott open set” for the “Scott Topology”

I am trying to understand the Scott Topology. I am looking at the page here: http://planetmath.org/scotttopology and for the definition of "upper set": http://planetmath.org/node/37801 I am having ...
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1answer
588 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...
3
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1answer
1k views

The product of Hausdorff spaces is Hausdorff

I'm confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, ...
3
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1answer
132 views

On continuously uniquely geodesic space II

This question was inspired by this answer of @wspin. Definition : A continuously uniquely geodesic space is a uniquely geodesic space whose geodesics vary continuously with endpoints. Question ...
3
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1answer
269 views

Prokhorov's theorem for finite signed measures?

Prokhorov theorem provides a useful characterization of relatively compact sets w.r.t. narrow topology (topology induced by narrow convergence) in the space of probability measure. Notation used ...
3
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1answer
174 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. ...
3
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1answer
299 views

Rudin's PMA Exercise 2.18 - Perfect Sets [duplicate]

I've been working through Chapter 2 questions and have thought about Exercise 2.18 for a while, but couldn't come up with an answer. Is there a nonempty perfect set in R which contains no rational ...
3
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1answer
247 views

A functional structure on the graph of the absolute value function

Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the graph of the absolute value function. That is, $X=\{(x,|x|) : x\in\mathbb{R})\}$. We define a functional structure on $X$ by restricting ...
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4answers
427 views

continuous onto map from $(0,1)\to (0,1]$

I need to know whether There exists any continuous onto map from $(0,1)\to (0,1]$ could any one give me any hint?
3
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2answers
126 views

Products and Stone-Čech compactification

Let $X$, $Y$ be complete metrizable spaces, $\beta X$, $\beta Y$ be their Stone-Čech compactifications. It is known that $C_b(X) \simeq C(\beta X)$. Is it possible to say something about the relation ...
3
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1answer
263 views

Characterization of the Subsets of Euclidean Space which are Homeomorphic to the Space Itself

I have no real experience in topology (although I have done a course in metric spaces) but in the course of a project I am doing it has become useful to produce (if possible) a characterization of the ...
3
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1answer
46 views

Brouwer transformation plane theorem

Can somebody show that BPTT, version 2 is deduced from BPTT, version 1 [BPTT, version 1] Let $h$ be a fixed point free orientation preserving homeomorphism of $\mathbb{R}^2$ Every point ...
3
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2answers
141 views

Two equal functions on a topological space

Can anybody help please help me, I have to answer this problem in topology: "Let $f$ and $g$ be continuous functions from the topological space $T$ into $\mathbb{R}$, with the usual topology. Show ...
3
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1answer
160 views

Question on the proof of a subspace of Polish space is Polish, iff it's a $G_\delta$ set.

Suppose, $X$ is a Polish space, $Y$ is a Polish subspace of $X$. $\{U_n\}_{n \in \Bbb N}$ is a basis of open sets of $X$. Let $A = \{ x \in \overline {Y} : \forall \epsilon \exists {n}(x \in U_n ...
3
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1answer
70 views

Isometries of [0,1] with natural topology

I'm trying to prove that metric space $([0,1], d)$ with natural topology has only one isometry other that identity. The hint in my book says that I should consider $f: [0,1] \rightarrow [0,1]$ such ...
3
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1answer
91 views

Prove that a connected space cannot have more than one dispersion points.

Prove that a connected space cannot have more than one dispersion points. I have no idea at all how can I able to tackle the problem.can anyone help me please. Also can I get some examples of ...
3
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2answers
264 views

Dieudonné complete and topologically complete are equivalent for every space $X$.

How can we show that: For every topological space $X$ the following conditions are equivalent: A space $X$ is topologically complete if $X$ is homeomorphic to a closed subspace of a product ...
3
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1answer
180 views

Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
3
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1answer
154 views

Is every $T_4$ topological space divisible?

A topological space $(X,\mathcal T)$ is said to be divisible iff for each neighborhood $U$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ in $X\times X$, there is a symmetric neighborhood $V$ of the ...
3
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1answer
80 views

Topological vector space locally convex

Hello how to show the following: Let $X$ be a vector space. Then a norm on $X$ induces a topology under which $X$ is a locally convex topological vector space. Moreover, let $D$ be a nonempty ...
3
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1answer
2k views

homeomorphism between the real projective line and a circle

I'm currently following an introductory course in geometry and it was mentioned that the real projective line is homeomorphic to a circle. Could someone please state the topologies on both the real ...
3
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1answer
159 views

Closure of open set in a dense subspace of topological space.

Let $Y$ is a dense subspace of topological space $X$ and $U \mathop \subset \limits^{open} Y$. My purpose is to show that $Cl_{Y}(U)= Cl_{X}(V)\cap Y$. and i have two Question; Is it always true ...
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1answer
310 views

Topological properties of symmetric positive definite matrices

Let $S$ be the set of all symmetric positive definite matrices of size $n\times n$. Which of the following statements are true? (a) $S$ is closed in $\mathbb{M}_n(\mathbb{R})$. (b) $S$ is ...
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1answer
730 views

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

I think the title of the question says it all. I unfortunately did not seem to conclude anything. Some ideas I had: It is easy to show that (given $T$ is the rotation) $\{T^n(\theta)\}$ is a set of ...
3
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1answer
72 views

How to show any convergent sequence is strongly discrete in Hausdorff space?

Given a space $X$ and $C \subset X$, we say that $C$ is strongly discrete if there exists a disjoint family $\{U_x: x\in C\} $ of open sets in $X$ such that $x\in U_x$ for all $x\in C$. The question ...
3
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3answers
173 views

What is the negation of this statement?

Let $(K_n)$ be a sequence of sets. What is the negation of the following statement? For all $U$ open containing $x$, $U \cap K_n \neq \emptyset$ for all but finitely many $n$.
3
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1answer
777 views

Continuous function on metric space

I'm trying to show: Let $(X,d)$ be a metric space and let $A, B$ be nonempty subsets, which are also closed and disjoint. Let $\rho_A:X\to \mathbb{R}$ be such that $\rho_A=d(x,A)$ and $\rho_B:X\to ...
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2answers
2k views

Cantor set: Lebesgue measure and uncountability

I have to prove two things. First is that the Cantor set has a lebesgue measure of 0. If we regard the supersets $C_n$, where $C_0 = [0,1]$, $C_1 = [0,\frac{1}{3}] \cup [\frac{2}{3},1]$ and so on. ...
3
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2answers
2k views

Prove that B is a basis for a topology

Let $X = \{0,1,2,3,\ldots\}$ (the non-negative integers), let $$B_1 = \{\{n\} : n \in X \text{ and }n > 0\}= \{\{1\}, \{2\}, \{3\},\ldots\}$$ $$B_2 = \{Z \subset X : X \setminus Z = \{1,2,\ldots ...
3
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1answer
622 views

Nowhere differentiability of Space-filling curves?

In a homework assigment, we were given a certain recursive definition of a space-filling curve $f : [0,1] \mapsto [0,1]^2$ and asked to determine where it is differentiable. My intuition tells me that ...