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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4answers
42 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
17 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
23 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
11 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
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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
16 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
16 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
20 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 ...
2
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0answers
17 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
29 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
32 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 ...
1
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1answer
41 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
21 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
41 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
35 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
26 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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0answers
20 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
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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
32 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 ...
4
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1answer
53 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
97 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
67 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
70 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
44 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
56 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
29 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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0answers
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 ...
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2answers
52 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 $\mathcal{B}$ ?
2
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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 ...
3
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1answer
34 views

If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
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1answer
31 views

Convergence of filters in topological spaces [on hold]

I'm having quite some trouble proving the following: 1) Let $X$ be a topological space. If any filter on $X$ converges to any point $x$ $\in$ $X$, show that $X$ is endowed with the trivial topology ...
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0answers
82 views

If $\bar X$ is open, then $X=\bar X$. [on hold]

Let $X$ a metric space and suppose that $\bar X$ is open. Suppose that $\bar X\neq X$. Let $x\in \bar X\setminus X$. By definition of $\bar X$ there is a sequence $(x_n)_{n\in\mathbb N}$ in $X$ such ...
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3answers
46 views

Is $E$ path connected $\implies \overline{E}$ connected?

Let $E\subset \mathbb R^n$ a path connectedness open set. Is $\overline{E}$ connected ? (where $\overline{E}$ is the closure of $E$). I tried to prove that it's true, but I don't get anything, may be ...
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0answers
37 views

Topology of metric space

If $C[a,b]$ is the set of all real valued continuous functions defined on $[a,b]$ and $(C, d)$ is a metric space where $d(x,y)=\max | x(t)-y(t)|$ and $t$ belongs to $[a,b]$. Then how can i determine ...
1
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2answers
22 views

$M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$

Is it true? $X= Y \cup Z$, $M$ is a subset of $Y \cap Z$. Suppose that $M$ is open in $Y$ and $M$ is open in $Z$ then $M$ is open in $X$.
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1answer
46 views

Topology of Metric Spaces

Why is the open interval $(-\infty,+\infty)$ not an open sphere with usual metric? We can find a radius such that the open sphere is subset of real line same as we find that for any open interval. ...
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2answers
68 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
3
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3answers
66 views

Trying to Understand a Remark about Zariski Topology

I'm reading some notes in which following remark is given: The Zariski topology is quite different from the usual ones. For example, on affine space $ \mathbb A^n$ a closed subset that is not ...
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0answers
25 views

Quotient Space $X^*$ is homeomorphic to the Subspace $S^2$ of $\mathbb R^3$

Let $X$ be the closed unit ball $\{ x^2 + y^2 \leq 1 \}$ in $\mathbb R^2$ and let $X^*$ be the partition of $X$ consisitingof all the one point set $\{ x \times y \}$ for which $x^2 + y^2 < ...
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2answers
42 views

Does this theorem for bases also hold for subbases?

Assume that we have a toological space $X$ with toplogy $\mathcal{T}$. If Y is a subspace of X, then $\mathcal{T}_Y=\{Y\cap U|U \in \mathcal{T}\}$ is a topology on Y (that it really is a topology, ...
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1answer
157 views

Topological idea of orientability of manifold

While reading Poincare Duality a new idea of orientability of manifold came in my mind.I dont know wheather this idea is new or not, or even true or false. My idea is following... A $n$-dim manifold ...
2
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2answers
49 views

Does this set tend towards a disc?

Let $p$ be a complex polynomial \begin{gather*} p:\mathbb{C}\longrightarrow\mathbb{C},\\ \deg p = n,\quad n\in\mathbb{N}. \end{gather*} Define the set $\mathcal{R}=\{z\in\mathbb{C}:|p(z)|\leq R\}$, ...
3
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0answers
51 views

Homeomorphism definition: why $f^{-1}$ and not another function?

If you have two topological spaces $X$, $Y$ and two continuous bijections $$g: X \to Y $$ $$f : Y \to X $$ then are $X$ and $Y$ homeomorphic? If not, is there a reason why the above does not serve as ...
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0answers
48 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
0
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1answer
45 views

Every convex set in $\mathbb R^n$ has a countable and dense subset?

Assume the space is Euclidean space. Why every convex set has a countable and dense subset? How about in metric space? Any ideas or references? It is used in process of proving Debreu's Theorem in ...
3
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2answers
50 views

Is the graph of $xy=1$ in $\mathbb C^{2}$ connected?

The graph of $xy=1$ in $\mathbb C^{2}$ is set of points $(x+iy,u+iv)$ that satisfies $$xu-yv=1$$ and $$uy+xv=0$$ How to find if this set is connected or not . I also have another ...
2
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0answers
39 views

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on $R$

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on Spec(R). I've just learned about Zariski topology so I really don't have ...