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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2
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1answer
117 views

$[0,1]^{[0,1]}$ is separable

This is from Dudley´s book: Let $I:=[0,1]$ with the usual topology. Let $I^I$the set of all the functions from $I$ to $I$ with the product topology. a) $I^I$ is separable. Hint: Consider function ...
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0answers
20 views

Souslin space and functional

I have a question about Borel $\sigma$-algebra on a Souslin space. Let $E$ be a locally convex topological real vector space which is a Souslin space, that is, the continuous image of separable ...
3
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1answer
58 views

Exercise 3.20 in Hartshorne on dimension of integral schemes of finite type over a field.

While working on exercise 3.20 in Hartshorne I've gotten stuck on something for a while. Given an integral scheme $X$ of finite type over a field I'm tring to show the existence of some affine open ...
3
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2answers
81 views

Any relations between the weak topology on a Banach Space and the weak topology on CW complexes?

I'm learning about CW complexes, which we'll say are topological spaces $X$ that admit a filtration $\emptyset \subset X^0 \subset X^1 \subset \cdots \subset X^n \subset \cdots$, with $X=\bigcup X^n$, ...
0
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1answer
35 views

Find $\text{int,cl},\partial A$ for $A\subseteq \mathbb{R}_{\ell}$ (sorgenfrey line)

a. Find $\text{int A,cl A and}\partial A$ for $A=[2,5]\cup\{-\frac 3 n\mid n\in\mathbb{N}\}$ (Assuming $A \subseteq \mathbb{R}_{\ell}$). b. Find the connectedness component of the singleton ...
0
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2answers
62 views

Prove that A compact hausdorff space is generated by a weak topology of C(X,R)

Prove that A compact hausdorff space is a weak topology generated by C(X,R) ,by using C(X,R) separates points of X i.e for x not equal to y there exists a function f in C(X,R) s.t f(x) not equal to ...
0
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1answer
39 views

Does every non-compact bounded metric space support an equivalent metric in which it is unbounded?

Consider $X$ be an infinite set. Let $d$ be a non compact bounded metric on $X$. Can we define an unbounded metric $d'$ on $X$ such that both the metric spaces $(X,d)$ and $(X,d')$ give the same ...
0
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1answer
65 views

Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
0
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1answer
50 views

Measurability of a pointwise limit of measurable functions

Fellows. I'm trying to prove some measurability result and I figured out a solution using the following and now I wonder if this is actually true. Let $X$ be a topological space and $Y$ be a Polish ...
1
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1answer
73 views

For continuous functions, preimage of open set is open.

Let $f$ be a continuous function from a metric space $X$ into $Y$. If $V\subset Y$ and $V$ is open, then show that $f^{-1}(V)$ is open. The proofs I've seen of the fact that open sets have open ...
2
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1answer
186 views

Reference request: Analysis, Algebra and Topology - Same author(s)/publisher(s), progressive order

Is there anywhere I can acquire a collection of all Mathematical undergraduate textbooks by the same publishing author, or authors(so that they are similarly written) and can be completed in a logical ...
0
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3answers
130 views

Show that an “open square” is an open set: Show that {(x,y) in R2 such that -1<x<1 and -1<y<1. } is an open set.

How do I prove that an "open" square, centered in the origin is in fact an open set? I've already have this geometrical argument: Let $S$ denote the square. Suppose $(x,y) \in S$. Let $\delta = \min ...
2
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1answer
30 views

Period of a point $\tau \in \mathcal{H}$ (upper half plane) and elliptic points

This is probably a silly question but I'll ask it anyway. Here goes: Let $\Gamma$ be a congruence subgroup of SL$_2$($\mathbb{Z}$). To each point $\tau \in \mathcal{H}$ (where $\mathcal{H}$ is the ...
0
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2answers
38 views

$f: X \to Y$ with $Y$ a topological space, find topology on $X$ s.t $f$ is continuous

We are given a function $f: X \to Y$, and moreover, that $Y$ is a topological space. We are to determine if there is a topology on $X$ such that $f$ is continuous, and moreover, determine if this ...
0
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2answers
34 views

A Tychonov space non-homogeneous wrt neiborhood bases cardinality

Is there a Tychonov space $(X,\mathcal T)$ with $a,b\in X$ such that $a$ has a countable neighborhood basis while $b$ does not have any countable neighborhood bases?
2
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1answer
39 views

Proof of the Lebesgue Covering Theorem

Lebesgue Covering Theorem : Suppose $\rho =\{G_n\}$ is a covering of a compact subset $K$ of $\mathbb R^p$. There exists a positive number $\lambda$ such that if $x,y \in K$ and $|x-y| < \lambda,$ ...
3
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1answer
35 views

Every locally compact space is compactly generated

I am using the following definitions (from Wikipedia): A space $X$ is locally compact if every $x \in X$ has a compact neighborhood; A space $X$ is compactly generated if a subset $A \subseteq X$ is ...
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0answers
62 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 ...
-1
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1answer
30 views

Every singleton is the intersection of a decreasing sequence of basic open sets

If I have a nice (meaning Hausdorff, second-countable, and locally compact) space $E$, then I know that there is a countable base consisting of relatively compact sets $\mathcal{B}:=\{U_n : n\geq1\}$. ...
0
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1answer
75 views

Constructing all compact Hausdorff spaces from a generalised sequential-limit operator

Given a set $X$ and a partial function $\lim:X^{\mathbb{N}}\rightarrow X$, define for all $M\subset X$, $$\overline{M}=\{x\in X :\forall (x_n)_{\mathbb{N}}\in M^{\mathbb{N}}\,((x_n)_{\mathbb{N}},x)\in ...
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1answer
41 views

The metric identification of a pseudometric on $C(\mathbb{I})$

I have a pseudometric $\mu$ on $C(\mathbb{I})$ defined by $$\mu(f, g) = |f(x_0) - g(x_0)|.$$ I then take the metric identification of $(M, \mu)$ and am asked what familiar space this metric ...
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0answers
42 views

“Exemplary spaces” for the class of spaces satisfying separation axioms

The Sierpiński space, $S=\{0,1\}$ with the topology $\{ \{\}, \, \{1\}, \, \{0,1\}\}$, is a $T_{0}$ space. And in some sense, it is the "prototypical" $T_{0}$ space, for, if $(X,\tau)$ is a ...
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3answers
51 views

Some Questions from the proof of the result : The unit interval $\mathbb I = [0,1]$ is compact

The unit interval $\mathbb I = [0,1]$ is compact I was trying to understand the proof of the above result from my textbook which goes like as follows. However, I have a few questions in mind. Please ...
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0answers
45 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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1answer
34 views

Topology on completely regular space is weak topology.

Let $(X,T)$ be completely regular space, $C(X,\Bbb R)=\{f\mid f:X\to\Bbb R \text{ cont}\}$. Then show that the weak topology generated by $C(X,\Bbb R)$ is $(X,T)$. Clearly weak topo is subset of $T$. ...
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1answer
59 views

Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals

Prove that the Cantor set cannot be expressed as the union of a countable collection of closed intervals whereas it's complement can be expressed as the union of a countable collection of open ...
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1answer
23 views

Let X be any non-empty set, and prove that in the lattice of all topologies on X each chain has atmost one compact hausdorff topology as a member.

Let X be any non-empty set, and prove that in the lattice of all topologies on X each chain has atmost one compact hausdorff topology as a member. Here discrete topology will not work. Here what is ...
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3answers
58 views

Let $f$ be a function from a topological subspace $A$ of $X$ to a Hausdorff space $Y$.

Let $f$ be a function from a topological subspace $A$ of $X$ to a Hausdorff space $Y$. Then there exists at most one continuous extension from $\bar{A}$ (closure of $A$) to $Y$. Problem is if $X$ is ...
2
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2answers
133 views

Mary Ellen Rudin's proof that all metric space are paracompact

Given a metric space $(X,d)$, show that the space is paracompact. I have no idea where to begin on this, and the proofs of this I have seen have been difficult for me to understand. Can anyone offer a ...
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5answers
104 views

How to prove that $\mathbb R$ with usual metric and $\mathbb R$ with the discrete metric are not homeomorphic.

I know that two metric spaces are homeomorphic if there is a function from one to another such that $f$ is continuous, one-to-one, and onto. I know how to prove two metric spaces are homeomorphic. I ...
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2answers
33 views

How does one interpret functions of topological spaces?

Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is ...
0
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1answer
51 views

Every point of the cantor set $F$ is a cluster point of both $F$ and it's complement $F^c $

Every point of the cantor set $F$ is a cluster point of both $F$ and it's complement $ F^c$ Attempt: $(a)~~$ First proving that every point of $F$ is a cluster point of $F$. Suppose $x \in ...
3
votes
1answer
67 views

Can a plane be split into three connected sets so that $\epsilon$-neighbourhood of any point of any one set also contains points of two other sets?

Math SE. This question was a shower thought of mine. I tried to come up with an answer by twisting comb spaces and cantor sets, but to no avail. I was educated as experimental physicist, so I ...
0
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1answer
50 views

Open sets are not relatively compact

The following is a question about the answer given here: I have been trying to prove that if $X$ is an infinite dimensional Banach space and $O\subseteq X$ is an open set such that its closure ...
0
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0answers
32 views

Boundary of a holomorphic functions

Let $G ⊆ \mathbb{C}$ a bounded open connected set and let $f : \bar{G} → C$ a holomorphic function: Is this true? $$∂f(G) ⊆ f(∂G)$$ What I have is that $f$ is open.
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1answer
49 views

Image of boundary of continuous open function

Edit to be more clear: Let X and Y be topological spaces and $f:X→Y$ a continuous open map. Is it then true that $∂f(A)⊂f(∂A)$ for every open $A⊂X$ such that $∂A \neq \oslash$. thanks for your help ...
0
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2answers
29 views

How do you actually show that the complement of the set of condensation points of an uncountable set in $\mathbf R$ is at most countable?

I can show that the condensation points form a perfect set, but can't show this. This question may have duplicates in stackexchange but apparently has no detailed answer. Any solution will be ...
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2answers
72 views

Open and closed subset of $\mathbb{R}$

How to show that the only subsets of $\mathbb{R}$ which are simultaneously closed as well as open are $\emptyset$ and $\mathbb{R}$ itself. Can someone tell me how to go to the proof of it? I have ...
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0answers
26 views

Error ? A subset $A$ of $ \mathbb R^p$ S.T $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ is interior of $A$ and $A^-$ is closure of $A$

Can there be a subset $A$ of $ \mathbb R^p$ such that $A^o = \phi$ and $A^- = \mathbb R^p$ where $A^o$ refers to the interior of $A$ and $A^-$ refers to closure of $A$ Attempt: By definition : $(i) ...
0
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3answers
64 views

closure = union of the set and the set of limit points

Let $S \subseteq \mathbb{R^n}$ and denote the set of limit points of $S$ as $S'$. Show that $S \cup S' = \bar{S}$ where $\bar{S}$ is the closure. (i) I want to show that $\bar{S} \subseteq ...
2
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1answer
54 views

Are homeomorphic, Hausdorff topologies on a set equal?

Let $\mathcal T$ and $\mathcal S$ be two topologies on a set $X$ and $(X,\mathcal T)$ and $(X,\mathcal S)$ be homeomoric and compact and Hausdorff. Is $\mathcal S$ equal to $\mathcal T$? I know the ...
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3answers
62 views

If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself

If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself. Attempt: I apologize if this is too basic but I ...
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2answers
35 views

Why is the pointwise product of complex-valued continuous functions on a locally compact Hausdorff space continuous?

Let $X$ be a locally compact Hausdorff space and let $C(X) = \{ f \colon X \rightarrow \mathbb{C}\ \vert\ f \text{ is continuous on } X \}$. How do I show that $fg \colon X \rightarrow \mathbb{C}$ ...
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1answer
44 views

Use contraction mapping theorem to prove

Help! I am taking a math course, and I just can't figure out this proof: Let $\alpha,\beta\in R^n$, $a\in R$, and $A$ be an $n\times n$ nonsingular matrix. Use contraction mapping theorem to prove ...
2
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1answer
33 views

If $X$ is a LCHS and $K, O \subseteq X$ with $K$ cpt & $O$ open, then $\exists U$ open s.t. $K \subseteq U \subseteq \overline{U} \subseteq O$?

I'm having trouble fully understanding the proof of this statement. Suppose $X$ is a locally compact Hausdorff topological space. Then if $K$ is a compact subset of $X$ and $O$ is any open subset ...
3
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1answer
26 views

Well-ordered family of open sets.

Let $X$ be a second countable space. I $\textbf{A}$ is a family of open set well-ordered by inclusion prove that this family is numerable. I have the following idea: Let $\textbf{B}$ be a ...
0
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1answer
74 views

Proving that $F_{x,n} = \left\{y \in \mathbb R^p :|y-x| \leq \dfrac {1}{n}\right\}$ is contained in $G$

Let $G$ be an open subset of $\mathbb R^p$. Let $A$ be the subset of $G$ whose coordinates are all rational numbers. Then, show that For each $x$ in $A$, there is a smallest natural number ...
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1answer
42 views

Using uniform continuity to decompose a path in the complex plane

I have a homework problem from Conway's Functions of One Complex Variable that I am stuck on. The problem statement is as follows: Let $G$ be an open subset of $\mathbb{C}$ and let $P$ be a polygon ...
0
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2answers
116 views

Understanding a problem in Munkres

This problem is from Chapter 2, Section 16, number 5 in Munkres' Topology. This is not a homework problem, but I'm trying to complete all problems from the sections covered in class. Let $X$ and ...
0
votes
1answer
91 views

Every open sub set of $\mathbb R^p$ is the union of countable collection of closed sets

Every open sub set of $\mathbb R^p$ is the union of countable collection of closed sets My textbook gave me hints as follows : Let $G$ be an open subset of $\mathbb R^p$. Let $A$ be the subset of ...