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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1answer
15 views

n-dim volume of a scaled ball

Let $B \subset \mathbb{R}^n$ be the unit ball with respect to an arbitrary norm $\|.\|$ (e.g. $B=\{x \in \mathbb{R}^n:\|x\| \le 1 \}$). I read in a book that it is easy to show: $vol_n(\epsilon ...
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32 views

locally-compact Hausdorff space, equivalent, compact, continuous

Let $X$ be a locally-compact Hausdorff space and $f\in C(X)$. Show that die following statements are equivalent: a) For every $\epsilon>0$ is the set $\{x\in X:|f(x)|\geq\epsilon\}$ compact b) ...
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2answers
66 views

Index of a Jordan curve

Winding number theorem: If $J\subset \mathbb{C}$ is a Jordan curve and a point $z$ lies in its interior domain, then the winding number $n(J,z)=\pm 1$. Now suppose that $J$ is smooth and we have the ...
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42 views

Find a graduate/research-level math tutor [closed]

I'm coming from the social sciences and decided about 5 years ago to research the topic of artificial intelligence. Having mastered the basics, I would now like to design some experiments of my own, ...
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1answer
30 views

How to show that if $X$ is Hausdorff and $ \big\{ (x, y) : x \sim y \big\} \subseteq X \times X$ is closed then $Y$ is Hausdorff?

Let $\sim$ be an equivalence relation on a topological space $X$, and let $Y = X/\sim$ be equipped with the quotient topology. How to show that if $X$ is Hausdorff and the set $\big\{ (x, y) : x \sim ...
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2answers
35 views

Show that $Q:=\{[a,b)\times[c,d): a,b,c,d \in \mathbb{R}; a < b,c < d\}$ is base of a topology and show that belonging topology is separable

I have to show that $Q:=\{[a,b)\times[c,d): a,b,c,d \in \mathbb{R}; a < b,c < d\}$ is the base of a topology in $\mathbb{R}^2$. As far as I know that means I have to show that $Q$ is closed ...
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0answers
28 views

compactness of a sequence space

Sorry if this question might be not well-posed, I'm very very new to topology. I have a compact set $S$ of sequences $(x_n)_{n\in\mathbb N}$ in $\mathbb R^n$ and those sequences are bounded, in the ...
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0answers
45 views

Show that a Set is dense in $(C[0,1],|| \cdot| |_\infty)$ [duplicate]

I have to show that the set: $$M :=\{f\in C[0,1]:\exists L \gt 0 \: \forall x,y \in [0,1] \space \space \space |f(x)-f(y)| \leq L|x-y| \} $$ is dense in $(C[0,1],|| \cdot| |_\infty)$ Any ideas? ...
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0answers
21 views

$X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does $X$ closed implies/if $S$ is closed?

Let $X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does the closed-ness of any one of $S$ or $X$ implies that the other set is also closed ?
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1answer
41 views

“Continuous maps are those maps that do not tear space apart”

In a tutorial I wanted to give a quick explanation of the property of continuity. One of the common intuitions for continuity is that it preserves connection: Continuous maps do not map connected ...
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1answer
14 views

Space admitting an irreducible connected open covering is irreducible

Let $\{U_i:i \in I\}$ be an open covering of topological space $X$, where $U_i \cap U_j \neq \varnothing$ for every $i,j$. If $U_i$ is irreducible for all $i \in I$ then $X$ is irreducible. I am ...
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0answers
38 views

Subspace of topological space has lower dimension

The dimension of a topological space $X$ is defined to be the length of the maximal chain of closed irreducible subsets $\varnothing \neq X_0 \subsetneq X_1 \subsetneq ... \subsetneq X_n \subset X$. ...
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1answer
25 views

$X$ hausdorff and $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ is closed implies quotient map is open.

Let $∼$ be an equivalence relation on a topological space X. $\ Y = X/∼ $ equipped with the quotient topology. How to show that if X is Hausdorff and the set $ \big\{ (x, y) : x ∼ y \big\} ⊆ X × X$ ...
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1answer
39 views

An introduction to torus and a related fundamental questions on quotient mappings

Let us consider the following mapping on the square [0,1] $\ X $ [0,1] as follow : ($\ x $,0)~($\ x $,1) and (0,$\ x $) ~ (1 ,$\ x $) . The quotient space defined on it is called the two ...
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1answer
14 views

connectedness in terms of open partitions of spaces

A space is usually defined to be connected if it can't be nontrivially partitioned into opens. Can we relax the word 'opens' to 'spaces' and equality to homeomorphism? In other words, is the ...
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1answer
29 views

Compact subsets in Topology of pointwise convergence

First of all, I know a similar question has been asked here compactness in topology of pointwise convergence, but I am still do not know how to identify compact subsets. Given a set $X$, endowed with ...
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3answers
63 views

What is the Euclidean topology on $\mathbb{R}^0$ like?

I am trying to prove that a topological space $(X,\mathscr{T})$ is a $0$-manifold if and only if it is a countable discrete space. In the process I have to show that there exist a homeomorphism from a ...
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1answer
20 views

Excercise 1(b) on Zero-dimensional Homology in Munkres

If $\phi:C_0(K)\to \mathbb{Z}$ is an epimorphism such that $\phi\circ \partial_1=0$ then show that $$H_0(K)\cong \frac{ker\phi}{im\partial_1}\oplus\mathbb{Z}.$$ My working is since $C_0(K)$ is ...
3
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1answer
171 views

The boundary of set is the set itself examples

Could you provide me some examples of sets, which are not based on Cantor's construction, that satisfy the property $\partial A=A$, that is the boundary of a set is the set itself?
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2answers
90 views

Show that two metrics known not to be strongly equivalent actually induce the same topology.

Suppose on $\mathbb{R}$, we have the usual Euclidean metric, $\rho_{1}(x,y) = \Vert x-y \Vert$, and also the metric $\rho_{2}(x,y) = \displaystyle \frac{\rho_{1}(x,y)}{1+\rho_{1}(x,y)}$. I need to ...
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2answers
66 views

Why Munkres §26 Exercise 11 is nontrivial?

This is probably a silly question, but I have a trivial (most likely wrong) reading of Munkres §26 Exercise 11: Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed ...
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1answer
48 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something ...
2
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1answer
30 views

Weierstrass Approximation Theorem for a Product Space?

I am faced with the following problem: Let $X$ and $Y$ be compact Hausdorff spaces and $f$ belong to $C(X \times Y)$. Show that for each $\epsilon > 0$, there are functions $f_{1}, f_{2}, \cdots , ...
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1answer
14 views

Under what assumptions on φ is Tco-φ a topology

Fix a set X, and let φ be a property that subsets A of X can have. Define Tco-φ = {U ⊆ X : A = ∅, or X \ U has φ } . Under what assumptions on φ is Tco-φ a topology on X? What I think: 1. X\X has φ ...
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1answer
26 views

Question about proof structuring.

Let $d$ and $d_1$ be metrics on $X$ and $Y$ and $T$ and $T_1$ be induced topologies on $X$ and $Y$, respectively. $f:(X,T) \to (Y,T_1)$ is continuous iff for each $x_0$ and $\epsilon > 0$, there ...
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3answers
6k views

Union of closure of sets is the closure of the union: true for finite, false for infinite unions

Let $A_i$ be a subset of a metric space for each $i\in \mathbb{N}$. Let $B_n := \bigcup_{i=1}^n A_i$. Prove (for any) $n \in \mathbb{N}$ that $\overline{B_n} = \bigcup_{i=1}^n ...
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0answers
46 views

Possible Generalization of a Manifold

A manifold $M$ is a second-countable, Hausdorff, locally Euclidean topological space. Obviously, there are advantages to requiring $M$ to be locally Euclidean, i.e. in some cases this allows $M$ to be ...
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1answer
35 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
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2answers
36 views

Intuition of product spaces

So I have a product space of the form: $X=X_1 \times \ldots \times X_n$ and I take two elements of it, say $x=\{x_1,\ldots,x_n\}$ and $x'=\{x_1',\ldots,x_n'\}$. Now suppose I take the following ...
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1answer
17 views

$cl(\cup A_i) \supseteq \cup cl(A_i)$ proof and example? [closed]

While attempting the proof I though it would be possible that $x \in \cup A_i$ and hence its neighborhood intersect $\cup A_i$ and hence $A_i$ but this $A_i$ keeps changing so as to deny the property ...
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1answer
57 views

Set-Theory . What does $\min( |N_1(i)|, |N_1(j)| )$ mean?

I'm getting deeper into graph-theory and I've been looking at different ways to visualize graphs using matrices along. Of course there is an adjacency matrix (A) where the column (j) & row (i) ...
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0answers
22 views

A question about punctiform point sets

Let E be a finite dimensional Euclidean space whose dimension is at least two and let P be a punctiform subset of E. Is the complement of P in E always connected and locally connected? I know that the ...
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2answers
66 views

Is a nonempty intersection of a collection of closed sets closed? [closed]

My intuition is yes, but how would you prove it? Would the nonempty intersection of a collection of closed sets always be closed?
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3answers
765 views

Is there a $G_\delta$ set with Positive Measure and Empty Interior?

It is like in the title. $G_\delta\subset\mathbb{R}^n$ with Lebesgue measure. Thanks for any help
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1answer
26 views

Cl(A union B) = Cl(A) union Cl(B) counter example where Cl stands for closure?

In my topology book I was asked to prove the above result but instead I think I found a counter example. Consider: X = {a,b,c,d,e} A = {a,b} B = {c,d} and Basis BS = {(a,b,e) , (b,e) , (c,e) , ...
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0answers
10 views

A simplicial circle as the link of a vertex in a polygon with side identifications.

The bold lines which make the link of $v$ aren't even closed polygons how can they be simplicial circles- if it means when you put all three together- if you label the vertices of the links 1,2,3,4 ...
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1answer
27 views

Why are uncountable discrete spaces never separable?

I have to show: $A:=\{(x,-x)\in \Bbb R^2\}$ is, as a Subspace of $(\Bbb R^2, T)$, discrete and therefore not separable. I've found the argument that since the boundary of $A$ is equal to $A$ ...
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3answers
47 views

If a set is open in one metric it is open in another?

Ive been struggling to grasp a certain situation involving metric spaces and was wondering if anyone could be of any help. In the notes for my module on metric spaces I have the following "If two ...
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2answers
78 views

Is it weird to say $A \in B \in C$?

I've just noticed that I've never seen any text say $A \in B \in C$, which is why when writing it myself it immediately looked weird. For context, I was proving a result about the topology ...
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3answers
2k views

Proof that a discrete metric is indeed a metric space

QUESTION Let X be any set and $d : X \times X \to \mathbf{R}$ be given by $$ d(x,y) = \begin{cases} 0, & \text{if $x = y$} \\ 1, & \text{if $x \neq y$} \\ \end{cases}$$ Show that $d$ is a ...
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1answer
14 views

Characterization of upper-semicontinuous mapping

Iam looking for a proof of the following assertion: Let $X$ be a real reflexive Banach space, let $T\colon X\rightrightarrows X'$ a point-to-set mapping such that for every $x\in X$ the set $T(x)$ ...
4
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1answer
52 views

If two matrices are path connected, so are their inverses

The set of $n\times n$ matrices can be identified with the space $\mathbb{R}^{n\times n}$. Let $G \le GL_n(\mathbb{R})$. We say that $A \in G$ and $B \in G$ are path-connected (not sure if this is ...
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1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
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1answer
28 views

A discrete metric space is complete

We can read here that every discrete metric space (where the topology is the same as the discrete topology, i.e. where all the singletons are open) is complete, but an example bothers me because I ...
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1answer
42 views

Finding measure of Cantor set

Let us consider the following function designed to measure the length of subsets of $\mathbb R$: $l^∗ : 2^{\mathbb R} → [0,+∞]$ satisfies $l^∗(∅) = 0$, $l^∗(S) \leq l^∗(T)$, if $S ⊆ T$, ...
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0answers
25 views

Homology and triangulation of open surfaces

For example I have an open disk, or an open annulus. How do I triangulate open surfaces to find their (simplicial) Homology? Well, I know that open disk and closed disk are both homotopic to a ...
2
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1answer
28 views

Some basic question on pasting map from a square to a Klein bottle and homology

Consider a square $S$ which edges identified as follows Let $K$ be a Klein bottle and $p:S\to K$ be pasting map. Let $X$ be the image of the interior of $S$ under $p$ and let $Y$ be the image of a ...
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2answers
33 views

Compactness implies closedness in $\mathbb R^n$

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\cl}[1]{\overline{#1}}$ $\newcommand{\e}{\varepsilon}$ I am showing from first principles that compactness in $\mathbb R^n$ implies closed-ness. The ...
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0answers
34 views

Relative Homology (Question about Example 2 in Munkres)

I have no problems for $p=0$ case and for $p\geq 2$ it is quite obvious since $C_p(K,v)=C_p(K),\forall p\geq 2$. Now the tricky is for $p=1$. Since the elements of the kernel now not anly map to $0$ ...
1
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1answer
32 views

The image of an injective function whose domain is a topological space also a topology

Let $(X, T )$ be a topological space, and let $f : X → Y$ be an injective (but not necessarily surjective) function. QUESTIONS. (1) Is $T_f := \{ f(U) : U ∈ T \}$ necessarily a topology on $Y$ ? ...