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
22 views

Closure in a topological product: is AC needed?

I'm working on a proof of $\prod_{\alpha\in\Lambda}\overline{A_\alpha}=\overline{\prod_{\alpha\in\Lambda}A_\alpha}$ in the product topology. This has been asked before, i.e. Closure in a product of ...
6
votes
5answers
53 views

Which of the following condition ensure that the function $f:R^n\to R$ is continuous?

I encountered an interesting problem in my Economics class about continuity. Which of the following conditions on the function $f:\mathbb R^n\to \mathbb R$ ensures that the function $f$ is ...
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0answers
5 views

equivalent definitions of locally compact space

Let $X$ be Hausdorff space. Equivalent: 1.every point of X has a compact neighbourhood. 2.every point of X has a local base of compact neighbourhoods. the direction $2.\Rightarrow 1.$ is clear. I ...
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2answers
22 views

Interior points are limit points in $\mathbb{R}$?

I have read another question, and know that interior points are not limit points in general topology space. But when we talk about any subset $\mathbb{A}$ of $\mathbb{R}$, can I say that ...
5
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1answer
63 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
2
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3answers
46 views

Boundary of a bounded open set in $\mathbb{R}^2$

Does the boundary of a bounded open set in $\mathbb{R}^2$ necessarily have infinite points? How do we prove that, or is there a counterexample? It seems true to me, but I haven't been able to find a ...
7
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0answers
30 views

Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
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0answers
24 views

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$} [on hold]

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$}
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2answers
30 views

Question about product topology notation

Instead of using the general form, I will use a simpler one such as $\mathbb{R} \times \mathbb{R}$ (which is $\mathbb{R}^2$ of course). Now the notation says that the open sets are the union of the ...
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2answers
21 views

Show that the projection map $p: \mathbb{R}^2 \to \mathbb{R}$, where $p(x,y) = x$, is open

I have to take an open set in $\mathbb{R}^2$ and show that it maps to an open set in $\mathbb{R}$. So let $A \times B$ be an open set in $\mathbb{R}^2$. I have to show that $A$ is an open set. By ...
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0answers
25 views

On classification of directed topological spaces [on hold]

Is classification of directed topological spaces (not their homotopy equivalence classes!) an important subject in modern mathematics?
7
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4answers
106 views

The Galois connection between topological closure and topological interior

[Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.] Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and ...
2
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1answer
43 views

Understanding Rudin's proof that compact subsets of metric spaces are closed.

Rudin's Principles of Mathematical Analysis has the following definition of compact: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. ...
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0answers
29 views

Can a linear projection of spheres be a torus?

Assume that we have two disjoint subsets $A_1, A_2 \in \mathbb{RP}^4$ that are both homeomorphic to the sphere $S^2$. Let $\pi$ be the linear projection with centre a point that does not lie on $A_1$ ...
1
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1answer
24 views

Finding an element in $l_1$ space with certain properties

I am facing a bit problem in the following: Given $x_1,...,x_m \in l^\infty$ and positive $\epsilon_1,...,\epsilon_m$, I need to find an element $a= (a_n)$ in $l_1$ space such that $\sum_{n=1}^ \infty ...
1
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2answers
43 views

$f: \mathbb{R} \to S^1$ where $f(x) = (\cos x, \sin x)$ open and closed mapping?

Show that $f: \mathbb{R} \to S^1$ where $f(x) = (\cos x, \sin x)$ is both an open and closed mapping, or provide counter-examples if one or both are not true. Well, my hypothesis is that they are ...
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0answers
51 views

rudin's definition of a compact set

Here are some definitions given in my book: Definition 2.31 By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha \}$ of open subsets of $X$ such that $E \subset ...
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2answers
25 views

Can we classify all spaces which go by the given below problem

In chat I was discussing this problem which I thought of while doing my revision: If $M$ is a subspace of the space $X$ and we have a mapping of $M$ from the space $Y$ can I extend this map to a ...
1
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1answer
29 views

Any homeomorphism from $D^2$ to $D^2$ maps $\partial D^2$ onto $\partial D^2$

I'm starting to study Algebraic Topology. After doing some problems and studying the theory I've arrived at: Let $D^2$ be the unit disk in $R^2$, $\partial D^2$ the topological boundary of $D^2$ ...
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7answers
481 views

What does it REALLY mean for a metric space to be compact? [duplicate]

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ...
0
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1answer
24 views

Question about “subset of topological space”

In the Topology book I'm studying, there is a exercise that starts off with the statement: Suppose $X$ is a topological space, $A$ is a subset of $X$... I'm not 100% sure what this means. First ...
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1answer
30 views

Show the “clock”and Euclidean metrics generate different topologies

I'm trying to teach my self topology. I wanted to find an example of a metric generating different topology. I came up with what a call "clock" metric, inspired by the modulo operation. Can anyone ...
2
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0answers
23 views

Int M is open and a manifold

If M is an n-dimensional manifold with boundary, then Int M is an open subset of M , which is itself an n-dimensional manifold without boundary. I am supposed to use these definitions: If M is an ...
0
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1answer
45 views

Give the example of compact set with infinite countable derived set [on hold]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
4
votes
1answer
57 views

Is the set open?

Define a complex polynomial $p:\mathbb{C}\longrightarrow\mathbb{C}$ where $\deg p=n\in\mathbb{N}$. \begin{equation} p(z) = \alpha_{n}z^{n}+\alpha_{n-1}z^{n-1}+\dots+\alpha_{1}z+\alpha_{0},\quad ...
1
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2answers
74 views

What's the disjoint union?

I'm self-studying some analysis, and ran into the term 'disjoint union'. I googled it, and it seems that it's just a normal union of any sets, but where we pair each duplicate with an index ...
0
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0answers
26 views

Proof that the Klein bottle can be immersed in $\mathbb{R}^3$ and embedded in $\mathbb{R}^4$

We define the Klein bottle as the quotient space of $I^2=[0,1]\times [0,1]$ under the relation $\sim$ for which $(0,y)\sim (1,1-y)$ and $(x,0)\sim (x,1)$. If we found a continous $f:I^2\to R^k$ which ...
4
votes
2answers
73 views

In $\Bbb C$, are polynomials open maps?

If $p$ is a polynomial, is it true that for every open $A\subseteq\Bbb C$, $p(A)$ is open? I really don't know how to approach this. I'm fairly certain that they're closed maps, though.
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0answers
44 views

Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
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3answers
24 views

Pre-image of $f(x,y) = xy$

$f: \mathbb{R^2} \to \mathbb{R}$ is $f(x,y) = xy$. Find the pre-image $f^{-1}((a,b))$ of an open interval $(a, b) \in \mathbb{R}$, and show that this pre-image is open in $\mathbb{R^2}$. I can't ...
3
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2answers
47 views

$f: X \to Y$ continuous if and only if $f: X \to f(X)$ continuous

Let the image of $f$, which is $f(X)$, be a subspace topology of $Y$. Prove that $f: X \to Y$ continuous if and only if $f: X \to f(X)$ continuous. 1) If $f: X \to Y$ continuous, then $f^{-1}(U)$ ...
2
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4answers
66 views

Why the proof of “compact set being closed in $\Bbb R$” fails for a non-Hausdorff space?

The following is a proof that "a compact set is closed in $\Bbb R$" from my real analysis course material. I just learned this is not so for a general topological space if that space is not Hausdorff. ...
0
votes
1answer
20 views

a question about how to find the closure of $(a;b)$ in the discrete topology

If $T$ is the discrete topology on the real numbers $\Bbb R$, find the closure of $(a;b)$. Is it $(a;b)$ or $[a;b]$? I just began to learn Topology, and felt a little confused over here.
2
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1answer
25 views

Understanding pasting lemma proof

Let $A$ and $B$ be both open or closed subsets of a topological space $X$ such that $A \cup B = X$. Let $f: A \to Y$ and $g: B \to Y$ be continuous such that $f = g$ for all $x \in A \cap B$. Prove ...
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0answers
14 views

Foliation vs Coordinates in de Sitter

I'm studying de Sitter manifolds and am confused about the difference between the choice of foliation and the choice of coordinates (and how they relate to the spatial curvature). I can choose the ...
2
votes
1answer
30 views

Quotient space is not second countable

I was searching for an easy example of a quotient space $X$/~ which is not a second countable space even though $X$ is a second countable topological space. I have found an example in the following ...
0
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1answer
18 views

Show $\mathcal N_x$ is neighbourhood system on X

I had following exercise : If we have a topological space $(X,\tau) ~~$and x$ \in X$ ,and suppose there is a family of sets defined as: $ \mathcal N_x=\{N_x;N_x\supset O_x \ni x,$for ...
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0answers
20 views

How to prove that $t\mapsto (\cos 2\pi t, \sin 2\pi t)$ induces the one-point compactification of $(0,1)$?

Take the unit circle $S^1 = \{(x, y) \in \mathbb{R^2}: x^2 + y^2 = 1\}$ and let $h: (0, 1) \to S^1$ be the map $h(t) = (\cos(2\pi t), \sin(2\pi t)$. The compactification induced by $h$ is the same ...
1
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1answer
21 views

A topology induced by neighbourhood system.

My topology teacher gave the following exercise: Let $X \neq\phi $ & $N_x$ be a neighbourhood system on $X$. Then take $\tau=$ those sets $O$ s.t. $O$ is neighbourhood of each point of ...
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0answers
22 views

Open subset of compactly generated space, compactly generated?

If each point of an open subset $U$ of a compactly generated $X$ has an open neighborhood in $X$ with closure contained in $U$, must $U$ be compactly generated?
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0answers
32 views

A counterexample that even all compacts are closed the topological space is not necessarily Hausdorff [duplicate]

One counterexample is the co-countable topology shown here. However last night we found the second solution in that post is incorrect by the discussion here. Can anyone help provide an alternative ...
2
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1answer
33 views

Boundedness in uniform spaces?

After looking a bit at uniform spaces, as their general definition seems relevant to the study of topological vector spaces, it seems that they provide just enough structure to define the notion of ...
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2answers
28 views

Cheap proof that the Sorgenfrey line is normal?

It is very easy to prove that the Sorgenfrey line is completely regular: To separate a point $x$ from a closed set $F$, note that there is an interval $[x,y)$ disjoint from $F$ and observe that ...
2
votes
1answer
50 views

Topological Version of First Isomorphism Theorem

Given a set $X$ and an equivalence relation $\sim$ on $X$, we can define the set $X_\sim=\left\lbrace\left[x\right]:x\in X\right\rbrace$ of equivalence classes, and we can define a projection map ...
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0answers
29 views

munkres topology the meaning of the uniform metric on $\mathbb{R}^X$

I've been going through Munkres' Topology on my own, and I've come across an exercise where I can't even understand the question, it's in section 21, number 7. Let $X$ be a set, and let $f_n: ...
2
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1answer
44 views

Help show all compact sets are closed in the compact complement topology

Given the usual topology $(\Bbb{R},\tau)$ on $\Bbb{R}$, define the compact complement topology as $\tau'=\{A\subseteq \Bbb{R}:A^C$ is compact in $\Bbb{R}\} \bigcup \{\emptyset \}$. The question is to ...
2
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1answer
37 views

Is any closed subspace of a $k$-space a $k$-space?

See here for a definition of $k$-space. As the title suggests, is any closed subspace of a $k$-space necessarily a $k$-space?
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1answer
32 views

How do I compute the fundamental group of this space?

Let $X$ denote the real projective plane. How do I compute the fundamental group of the connected sum $X\#X$? I'd like to use Van Kampen's theorem, but I have trouble visualizing what this space ...
3
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0answers
24 views

Is this space second countable?

I have posted questions about this space before. I'm hoping someone can comment on if my logic on this space not being second countable is right. Consider the following space: the underlying set is ...
2
votes
1answer
20 views

Is any subspace of a weak Hausdorff space necessarily weak Hausdorff?

As the title suggests, is any subspace of a weak Hausdorff space necessarily weak Hausdorff. Thanks.