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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8 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 $A_1$ ...
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16 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 ...
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2answers
36 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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39 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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24 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 ...
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1answer
27 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
353 views

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

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 ...
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1answer
23 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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26 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 ...
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19 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 ...
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1answer
35 views

Give the example of compact set with infinite countable derived set

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 .
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1answer
55 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 ...
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2answers
71 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 ...
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0answers
23 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 ...
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2answers
72 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
33 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
22 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 ...
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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 $f^{-1}(U)$ ...
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4answers
62 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. ...
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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.
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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 ...
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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 ...
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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
18 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 ...
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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
21 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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31 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 ...
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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 ...
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1answer
45 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
28 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: ...
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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
28 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 ...
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22 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 ...
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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.
3
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0answers
33 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
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1answer
54 views

Sequence of topological spaces

A friend of mine did an exercise where a part of the text was: In $\mathbb{R}^3$, with euclidian topology, we consider $X=\mathbb{S}^2 \setminus \{ N \}$, where $N= (0,0,1)$ and $E=\{(x,y,z) \in ...
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3answers
99 views

Is $\mathbb R$ with usual topology a Hausdorff space?

$\mathbb R$ with usual topology is a Hausdorff space. By the definition of Hausdorff space, for any two distinct point in $\mathbb R$ we can find disjoint neighbourhoods... Now consider the set ...
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3answers
68 views

How to define a continuous map from $[0,1] $ onto/into $\mathbb R$? [on hold]

It is possible to define a continuous map from $[0,1]$ onto $\mathbb R$?
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2answers
73 views

Manifolds with a finite but not trivial fundamental group

I came across this nice result: Theorem: If $M$ is a connected smooth manifold with finite fundamental group, then its first de Rham cohomology is trivial: $$H^1_{dR}(M)=0.$$ However, I don't ...
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0answers
23 views

Topology generated by basis equals intersection of all topologies that contain A

Here is my proof I was wondering any critiques to my proof. If A is a basis;The topology generated by A equals collections of all unions of elements of A that is $\tau = \bigcup_{i \in I: B_i \in ...
2
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1answer
46 views

Why is the measure of a boundary of an open ball positive in only a countable number of cases?

Let $X$ be a Polish (complete separable metric) space and $\mathbb{P}$ a Borel probability measure on $X$. Let $x_1, x_2, \ldots$ be a sequence of points dense in $X$. How can you prove that there is ...
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1answer
57 views

Is every simply connected open subset of $\Bbb R^n$ contractible?

Question: Is every simply connected open subset of $\Bbb R^n$ contractible? I know the result is true for $\Bbb R^2$ because by the Riemann Mapping Theorem every simply-connected proper open ...
3
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0answers
33 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
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22 views

continuity of the piecewise functions [on hold]

$1$. $g(x)=0$,if $x$ is irrational and $g(x)=x$ if $x$ is rational Find all points of at which $f$ is continuous. $2$. Let $A$ and $B$ be compact sets. Define $A+B =$ ...
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1answer
27 views

Confused with order topology

What does $0\times 1$ mean in the order topology $?$ How does ${{1}\over{2}} \times 0$ look like? Are they just a point or a line$?$ How do i visualize them$?$ I understand that $[0,1]\times[0,1]$ is ...
4
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2answers
53 views

If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?

Let $\left(X,\tau\right)$ a topology space and $\mathcal{B}$ a base of the topology, my question is: The Borel $\sigma$-algebra is generated by $\mathcal{B}$ ?
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0answers
23 views

Find a set with empty interior and boundary equal to closure of $B^2$

I'm trying to find a set $A$ in $\mathbb{R}^2$ such that $\operatorname{Int}(A)$ is empty and $\operatorname{Fr}(A)=\operatorname{Cl}(B^2)$ I'm not sure how to do this. If I define my set as ...