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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2answers
19 views

Is the following text correct about the interior of a given set? (excerpt from Stephen Boyd's convex optimization text)

How is the interior of set C empty in this example? There is definitely more than one $x \in C$ such that $B(x,\epsilon) \subset C$.
2
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1answer
27 views

Proving equivalence of topologies using subbases

Suppose I have two topologies $\mathcal{T}$ and $\mathcal{T}'$ on a set $X$. Furthermore suppose $\mathcal{T}$ is generated by a collection $\mathcal{E} \subseteq \mathscr{P}(X)$ and $\mathcal{T}'$ ...
1
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2answers
40 views

Two definitions of Closure

Define the closure of a set as the intersection of all the closed sets that contain it. i.e; $$\mathrm{cl}(A) = \bigcap\{ C\mid A\subseteq C\quad\text{and}\quad C\text{ closed}\}.$$ Prove that the ...
1
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1answer
18 views

Proving the derived set $E'$ is closed.

I was reading the proof in Rudin, but it uses the metric. Is this not true if $X$ is a general topological space and $E' \subset X$ (especially if it is not Hausdroff $T_1$)? I can't come up with a ...
0
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1answer
26 views

Lebesgue Number Lemma

I am studying the Lebesgue Number Lemma: Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ having ...
1
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1answer
36 views

If a point is not isolated then it is a limit point

Let $k \in X$, where $X$ is a metric space. I want to show that, if $k$ is not an isolated point of $U$ where $U \subseteq X$, then it is a limit point of $u$ if $\exists u_{n} \in U$ whose elements ...
0
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1answer
35 views

Existence of two disjoint closed sets with zero infimal distance

Are there two closed sets $A,B\subset\mathbb{R}^2$ with the following properties? $A\cap B=\emptyset$ $\forall \epsilon>0$ there exist $a \in A$ and $b\in B$ such that $\|a-b\| < \epsilon$
1
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2answers
38 views

Completeness of closed and open balls in a metric space

Let $B_{r}(x) = \{y \in X \mid p(x,y) < r\}$ be an open ball and $\bar{B_{r}}(x) = \ [y \in X \mid p(x,y) \leq r\}$ be a closed ball. Why is it that a closed ball is a complete metric space while ...
2
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1answer
38 views

Equivalent topologies on Real projective space $RP^{n}$

This is homework,so no answers please. Prove that the topology on $RP^{n}$ given by the standard smooth structure ($\tau_{1}$) (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$) is equal to the ...
2
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1answer
24 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
0
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1answer
47 views

Is this proof correct $A=\bar{A} \implies A$ is closed

To prove $A=\bar{A} \implies A$ is closed In order to prove the above implication,it is sufficient to prove that i $A=\bar{A} \implies A^C$ is open meaning $A^C$ is an element of the topology Since ...
1
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1answer
60 views

Proving, that closure of set is equal this set iff set is closed

I've started intorduction to topology course and I need help with one of the problems: Let $A \subset(X,T). $ Prove that $cl(A) = A\iff A$ is closed. It may looks trivial, but I had a little ...
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1answer
28 views

prove that $F^{-1}$ is closed

Can you help me proving this lemma. That if $F : X \rightarrow Y$ ($X$ and $Y$ are topological spaces) is bijective and closed , then $F^{-1}$ is closed.
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1answer
27 views

Partition of Unity's Lemma

Let $V\subset\mathbb{R}^n$ compact, $\Omega\subset\mathbb{R}^n$ open, $V\subset\Omega$, $\delta:=\inf\{|x-y|\mid x\in V,y\notin\Omega\}$, $U:=\left\{x \mid |x-y|<\frac{\delta}{2}\,\,\text{for ...
1
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3answers
48 views

Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge

This is a homework problem for an undergrad topology course. Let $l^{2}$ be the set of all real-valued sequences $(c_{n})$ where $\sum_{n=1}^{\infty}c_{n}^{2}$ converges. Let $(a_{n}),(b_{n})\in ...
4
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1answer
38 views

Is the “product rule” for the boundary of a Cartesian product of closed sets an accident?

Given two closed sets $A$ and $B$ living in topological spaces $X$ and $Y$, the boundary of $A\times B$ in the product topology, denoted (suggestively) by $\partial(A\times B)$, is given by ...
-1
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2answers
30 views

Continuity of the union of two functions in a topological space

Let $X$ be a topological space with closed subsets $A$ and $B$ such that $X = A \cup B$. Let $f: A \rightarrow Y$ and $g: B \rightarrow Y$ be continuous functions such that for $x \in A \cap B$, ...
0
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1answer
46 views

prove that if for every Cauchy sequence $\{y_n\}$ there exists a Cauchy sequence $\{x_n\}$ such that $\dots$ [on hold]

let $(X,d)$ and $(Y,\rho)$ be complete metric spaces. let $f\space :\space X\rightarrow Y$. Prove the following: If for every Cauchy sequence $\{y_n\}\subset Y$ there exists a Cauchy sequence ...
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1answer
21 views

prove that for every $x\in X$ exists $n\in \mathbb N$ such that $interior(f(B[x,n]))\not = \emptyset$

let $(X,d)$ and $(Y,\rho)$ be complete metric spaces and let $f\space:\space X\rightarrow Y$ be a surjective closed function. prove that for every $x\in X$ exists $n\in \mathbb N$ such that ...
2
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3answers
34 views

$\mathbb{R}$ with the finite complement topology

Let $X=\mathbb{R}$ be given with the collection $\tau$ where $$ \tau = \{U\subset X: |X\setminus U|<\aleph_0\}\cup \{\emptyset\} $$ I was sitting at my computer, when I suddenly asked myself: "How ...
0
votes
1answer
28 views

Nulhomotopic map from $S^1 \rightarrow \mathbb{C} - \{0\}$

Hullo, I am aware that the inclusion map $i : S^1 \rightarrow \mathbb{C} - \{0\}$ is not nulhomotopic since there is a retraction from $\mathbb{C} - \{0\}$ to $S^1$ making the induced homomorphism ...
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2answers
36 views

Open set in subspace not open in the entire space example

I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X. My own approach was this: Suppose U is a subset ...
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2answers
25 views

Convergence and finer topology

Can convergent of sequence be used to determine which topology is finer(in general topological space). I am asking this is question in effect of theorem on metric space: 'topology 1 is finer than ...
0
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4answers
37 views

Explaining what is Pathwise-connectedness.

I'm an average guy but interested in explaining myself maths through illustrations and intuition(which at times fails!!!).I'm preparing myself for my calculus of several variables exam. I'm studying ...
2
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1answer
30 views

Convergence of sequence and interior points

For a subset $A \subseteq X$, consider the statement, "$x$ is an interior point of $A$ iff for every sequence $(x_m)$ in $X$ converging to $x$ there exists $n \in \mathbb{N}$ such that for all $m > ...
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3answers
22 views

Given a disconnected subset of $X$, make a continuous function on $X$ that has value 0 and 1 on each component of separation

Suppose $X$ is a topological space and $A\cup B$ is a disconnected subset of $X$ where $(A,B)$ is a separation of $A\cup B$. Does this imply the existence of a continuous function $f : X \rightarrow ...
0
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0answers
28 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 ...
2
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1answer
29 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
0
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1answer
19 views

Removing a point from a closed mapping.

Suppose $f:X\to Y$ is continuous and closed surjection. Suppose $x\in X$ and $y\in Y$ are such that $f^{-1} \{y\}=\{x\}$. Then $f\restriction _{X\setminus \{x\}}:X\setminus \{x\}\to Y\setminus ...
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0answers
42 views

P-norm Unit Ball

Proof that for $0<p<1$, $p\in \Bbb{R}$ $$\|(x,y)\|_p=(|x|^p+|y|^p)^{\frac{1}{p}}$$ doesn't define a norm in $\Bbb{R}^2$. However, $$d_p((x_1,x_2),(y_1,y_2))=\sum_{i=1}^2|x_i-y_i|^p$$ defines a ...
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0answers
32 views

Proof related to Tychonoff's Theorem [on hold]

A proof related to Tychonoff's Theorem is that the intersection of a finite number of members of the maximal class M also belongs to the class M.  (See for example problem 7 p175 in Lipschutz's ...
0
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1answer
30 views

X × Y \ {(x, y)} is path connected if X and Y are both path connected [on hold]

i was solving exercise questions and came across this problem on connectedness ...Let X and Y be path connected spaces and (x, y) ∈ X × Y . If each X and Y has more than one elements then X × Y \ {(x, ...
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2answers
26 views

tupules $(x,y)$ with at least one entry rational is connected in $R^2$

I have studied connectedness and came across a problem which goes like this.. all the tuples $(x,y)$ with at least one entry rational is connected in $\Bbb R^2$. I have tried to prove it by ...
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1answer
33 views

Equivalence between two topological statements concerning the basis of a topology.

I need to show the following statement Let $\mathcal{B}\subset P(X)$ be a set of subsets of a set $X$, such that $\bigcup_{U\in \mathcal{B}}U =X$ then the following are equivalent $i)$ there ...
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0answers
22 views

Lindelöf subspaces in the product of ordinal spaces [on hold]

Let $$X = W_1 \times ( W_1 + 1 ).$$ Where W1 is the first uncountable ordinal number . How can we describe all Lindelöf subspaces of X ?
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1answer
55 views

Continuity upon trivial topology

I am puzzled with the following statement: "Given any map $f:X\to Y$ where $X$ is equiped with the trivial topology $(\varnothing,X)$, then this map is continuous iff $Y$ has the trivial topology. ...
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1answer
45 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
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2answers
34 views

Proving that a set $A$ is dense in $M$ iff $A^c$ has empty interior

Prove that a set $A$ is dense in a metric space $(M,d)$ iff $A^c$ has empty interior. Attempt: I think I proved the converse correctly, but I'm not sure how to start the forward direction. ...
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0answers
14 views

how to do classification of topological space which a poset is a frame

is module in algebraic geometry for classification of topological space which a poset is a frame which invariant is for doing this classification of topological space? if want to do full combination ...
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0answers
23 views

Computing transition map of $S^2$.

First please have a look at the cruddy diagram I have drawn. (it is at angle because my camera casts a shadow if I photograph it from above) Define the coordinate charts that map a portion of the ...
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0answers
27 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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0answers
25 views

$\{X_\alpha\}_{\alpha\in J}$ family of connected spaces, $X$ a product space. Show $X$ is connected.

Let $\{X_\alpha\}_{\alpha\in J}$ be a family of connected spaces; let $X$ be the product space $$X=\prod_{\alpha\in J} X_\alpha .$$ Let **a**$=(a_\alpha)$ be a fixed point of $X$. (a) Given any ...
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1answer
31 views

Closure of a certain set

Let $X$ be the ordered square (i.e. $X= [0,1] \times [0,1]$). X is in the order topology. Let $A = \bigl\{\frac{1}{n}\times 1: n \in \mathbb{Z}_+\bigr\}$ a subset of X. Could you please give some ...
0
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1answer
47 views
+50

A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
2
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1answer
46 views

Comparing certain topologies

Consider the following topologies on $\mathbb{R}$ $\mathcal{T}_1=$ the standard topology $\mathcal{T}_2=$ the lower limit topology $\mathcal{T}_3=$ the topology having as basis all open rays $(- ...
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1answer
41 views

Homeomorphism - transforming mug into donut

I read that a map is 'visually' a homeomorphism if you don't have to fold or tear the object. Thus, I was wondering what the problem with folding is? I guess that in this statement they don't assume ...
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1answer
55 views

Is “connected, simply connected” Redundant?

Here are my definitions of "connected" and "simply connected." A topological space $X$ is connected if and only if it is not the union of two nonempty disjoint open sets. A topological space ...
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0answers
23 views

Interior of dense set in ultraregular space

Let $X$ be a regular Hausdorff topological space without isolated point. We say that $X$ is ultraregular if the following condition holds: for every subset $A \subset X$, such that $A$ and ...
2
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0answers
25 views

Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the ...
2
votes
1answer
45 views

Is every closed ball (or open ball) in the Eucledean Space $R^n$ convex?

I am solving a problem and I need to use this fact: Every closed ball (or open) in the Eucledean Space $R^n$ convex? Hoever, I am not sure if it is true or not. Can anyone help? Thanks!