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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Basis in topology and AC

Let S ⊂ ℘(X). Let T be the coarsest topology on X which contains S. Then we call T the topology generated by S. Let S ⊂ ℘(X). Then we can easily prove using (generalised distributive law) the ...
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28 views

A question about the interior of a set.

If $(X,d)$ is a metric space and $A \subseteq X$ then the $int(A)$ is the union of all open sets contained in $A$. Then we have that $int(A)$ consists of all interior points of $A$. Everytime I see ...
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10 views

Characterize the following sets as closed/open in the space of M2(R)

Characterize the following sets as closed/open in the space of $M_2(R)$(topologized by considering it as a subset of euclidean space of dimension $4$ in the obvious way ) Set of matrices of the ...
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29 views

Write the Interior and closure of the following Sets in the space $\{5\}\cup(0,2]$

Write the interior and closure of the following sets in the space $\{5\}\cup(0,2]$ $[0.5,2]$ $(0,0.5] $ $\{5\}\cup(0,0.5]$ I need step by step solution , I missed the class during illness, so ...
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1answer
129 views

Two point topological space

Is there a standard name for the two point space with precisely one singleton being the only nontrivial open set? What are its most noteworthy categorical properties?
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30 views

Almost-discrete topological spaces

Call a topological space almost-discrete iff it can be obtained from discrete topological spaces by any combination of: (small) products, (small) coproducts, subspaces, quotient spaces. (Infinite ...
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1answer
74 views

Showing a function on $\mathbb{R}^2$ is a surjective and continuous

Given an open set $U$ (in the standard topology on $\mathbb{R}^2$) and a function $f:U\rightarrow \mathbb{R}^2$, I would like to show that, assuming $f$ is injective (1 to 1) and continuous: 1) $f$ ...
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2answers
96 views

Is compactness a generalization of completeness

Is the concept of compact spaces a generalization of completeness to non-metric topological spaces?
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32 views

Textbooks for Algebra, Analysis, Combinatorics, Geometry and Topology :) [on hold]

I have the list of some of the areas of Mathematics from wiki: Algebra Analysis Combinatorics Geometry and Topology I want the best textbooks from all of these fields. My compilation: Algebra - ...
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2answers
17 views

Are lines in arbitrary normed vector spaces closed?

Let $(V, \| \cdot \|)$ be a normed (real) vector space. Given two vectors $a$ and $d$ (with $d$ not the zero vector), is the line $ L = \{a + td: t \in \mathbb{R}\} $ through $a$ in direction $d$ ...
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25 views

Conditions any dense embedding from $(0,1]$ into $[0,1]$ must satisfy

This is a proof-verification request. Suppose that $m:(0,1]\to[0,1]$ is a dense embedding. That is, $m$ is continuous; $m$ is injective; the image $m\big((0,1]\big)$ is dense in $[0,1]$; $m$ has a ...
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33 views

Cantor-set construction problem [on hold]

I'm having some trouble getting started with this proof. Any advice would be helpful. Thanks. Show that the set of numbers in the interval [0,1] having decimal expansions using only odd digits is ...
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2answers
32 views

The set of numbers of the form $k/5^n$ is dense in the real line [on hold]

Show that the set of numbers of the form $k/ 5^n$, where $k$ is an integer and $n$ is a positive integer, is dense in the line. I'm having a little trouble getting started on this problem. Any ...
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1answer
31 views

Let $X$ denote the subset of $\mathbb{R}^{\omega}$ such that $\sum (x_i)^2 $ converges…

Let $X$ denote the subset $\mathbb{R}^{\omega}$ consisting of all sequences $(x_1,x_2,...)$ such that $\displaystyle \sum x_i^2$ converges. a) Show that if $\mathbf{x,y} \in X$, then ...
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1answer
36 views

topology - limit point question

Let $(X,d)$ be a metric space and let $U \subseteq X$. I want to prove that $x \in X$ is a limit point of $U$ if $\forall \epsilon > 0, |U \cap B_{\epsilon}(x)| = \infty \Leftrightarrow \forall ...
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0answers
45 views

The metamathematics of Brouwer's fixed point theorem [on hold]

I have just been introduced to Brouwer's fixed point theorem. It seems plain that this result is saying something very fundamental about the nature of the continuum and not just about continuous ...
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23 views

Projection from triangle to $\mathbb{R}^2$.

I constructed the $2$-simplex as follows, $$\triangle^2= pe_1+qe_2+re_3 \hspace{4mm} p,q,r \in \mathbb{R}$$ I want to project this triangle down to $\mathbb{R}^2$, that is so I can write ...
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2answers
32 views

The set of all segments is convex

If $X,Y\subset \mathbb R^n$ are convex sets, is it true the union of all segments $[x,y]$, where $x\in X$ and $y\in Y$ is a convex set? I've drawn pictures and I convinced myself that this is true, ...
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1answer
33 views

Convergence of $a_n=d(u_n,\Bbb{Z})$ where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$.

I would like to study the sequence defined by $$ a_n=d(u_n,\Bbb{Z}). $$ Where $u_n=\bigl(\frac{1+\sqrt{5}}{2}\bigr)^n$ and $d(u_n,\Bbb{Z})=\inf\{d(u_n,x):x\in\Bbb{Z}\}$. I do not really ...
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2answers
30 views

Rigorous Covering Space Construction

Construct a simply connected covering space of the space $X \subset \mathbb{R}^3$ that is the union of a sphere and diameter. Okay, let's pretend for a moment that I've shown, using van Kampen's ...
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58 views

An argument with my friend over $\bigcup [a+\frac{1}{n},b]$

I read it somewhere that the interval $(a,b]$ is a $G_{\delta}$ set as well as a $F_{\sigma}$ set Well I quickly wrote down $(a,b]=\bigcap_{n=1}^{\infty} (a,b+\frac{1}{n})$ which makes it a ...
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2answers
22 views

Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
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2answers
30 views

Subsets of a set with trivial topology

Question: Let $X$ be any set with at least two elements. Assume that the only open subsets of $X$ are the empty set $\emptyset$ and $X$ itself. - Which subsets of $X$ are closed? - Which subsets of ...
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30 views

if $A$ is open in $M$ and $B$ is open in $N$, then $A \times B$ is open in $M \times N$

where $d((m_1,n_1),(m_2,n_2)) = d_M(m_1,m_2) + d_N(n_1,n_2)$ By some propositions, $A$ is open in $M$ if there exist an open set $K_1$ such that $A = M \cap K_1$ Also, there exist an open set $K_2$ ...
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31 views

Visualization of Gauss Bonnet geometric objects

Where can we get to see some individual surface/line combinations in isometry visualizations with constant $ \int k_g ds $ (say total tangential rotation) ? Or with constant integral curvature ...
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22 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 $B(x,\epsilon) \subset C$.
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1answer
29 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 $\mathcal{T}'$ ...
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2answers
49 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 ...
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1answer
24 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 ...
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1answer
28 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 ...
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1answer
38 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 ...
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1answer
36 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 $\|a-b\| < \epsilon$
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40 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 ...
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1answer
54 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 (lines through the origin in $\mathbb{R}^{n+1}/\{0\}$ and $\tau_{1}$) is equal to ...
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1answer
32 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 ...
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1answer
49 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 ...
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1answer
62 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
30 views

prove that $F^{-1}$ is closed [on hold]

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
28 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 ...
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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 ...
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1answer
41 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 ...
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1answer
32 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 $x \in A \cap B$, ...
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1answer
49 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
23 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 ...
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3answers
39 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 ...
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1answer
30 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
37 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 ...
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4answers
38 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 > ...