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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6 views

Meager subset in $\omega$

Suppose we do have a filter $\mathcal{F}$ on $\omega$ which contains the cofinite filter, so $X\in\mathcal{F}$ implies $X$ is infinite. For $X\in\mathcal{F}$, let $f_X$ be the increasing enumeration ...
4
votes
1answer
746 views

When is Quotient Map a Covering Map

Group $G$ acts on topological space $X$. Also, $x,x'\in X$ not in the same orbit of $G$ have open $U$, $U'$ such that $g(U)\cap U'=\varnothing$ for all $g\in G$. I have shown that $X/G$ is Hausdorff....
9
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1answer
314 views

Is there an abelian cat of topological groups?

There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So i'm wondering: Is there a "suitably well behaved"...
17
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1answer
448 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
3
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1answer
50 views

Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
3
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1answer
38 views

Cartesian closed subcategories of compact Hausdorff topological spaces?

The category of compact Hausdorff topological spaces is famously not cartesian closed. I was wondering how much more one has to assume to actually arrive to a cartesian closed category. For example, ...
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2answers
17 views

Prove that the product of a countable number of separable spaces is separable space.

Prove that the product of a countable number of separable spaces is separable space. If each $(X_i,T_i)$ is separable, let $A_i \subseteq X_i$ be a countably dense subset. Then $ cl(A_1 \times A_2 \...
5
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3answers
106 views

What are some fields that intersect topology and number theory?

I see that number theory is studied from the algebraic and analytics aspects, but I have not seen any approach from topology or axiomatic set theory (using them to investigate the properties or ...
2
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3answers
64 views

Prove that $[a,b]\cong[c,d].$

Let $a,b,c,d$ be real numbers with $a<b$ and $c<d$. Prove that $$[a,b]\cong[c,d].$$ I know that I need to show that $[a,b]\cong[0,1]$ and $[c,d]\cong [0,1]$ then, by the transitive propety of ...
-2
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0answers
21 views

For a compact metric space $X$, does the subset of surjective continuous maps $X\to X$ have non-empty interior? [on hold]

Let $(X,d)$ be a compact metric space and denote with $S(X)$ the set of all continuous maps $f:X\rightarrow X$. If $d_U(\varphi, \psi):=\sup_{x\in X}d(\varphi(x), \psi(x))$ for $\varphi, \psi \in S(...
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1answer
47 views

Is this a proof for Heine-Borel Theorem in $R$ and if it is why doesn't this work for open sets?

$[a,b]$ has an open cover then every point of this interval is covered in an open set. Since a is in an open set some neighborhood of a is a subset of that set. Say $[a,a+r_1) \subset O_i $ similarly $...
0
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1answer
38 views

Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
0
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1answer
44 views

Regular topological spaces need not to be normal

I was looking for a counterexample for the following statement: "A regular topological space need not to be normal." I don't understand how to use the lemma to prove Theorem 7: http://fac.hsu.edu/...
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1answer
24 views

How do I formalize the topology generated by a subbasis?

The topology generated by a subbasis $\mathcal{S}$ is defined as the colection $\tau$ of all unions of finite intersections of elements of $\mathcal{S}$. I want to formalize $\tau$ as something ...
3
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1answer
39 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
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votes
3answers
47 views

What is common notation for “disjoint union of copies of $\mathbb{R}$”?

I'm looking at a question out of Lee's Smooth Manifolds: Show that a disjoint union of uncountably many copies of $\Bbb{R}$ is locally Euclidian and Hausdorff but not second countable. My ...
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1answer
30 views

Convex Sets in Topology

In Munkres Topology, he uses the term 'A subset Y of X that is convex in X' in page 91. I couldn't find a definition for it inside the book and the definitions outside the book(on the internet) seems ...
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0answers
27 views

Riemann integrability question in $\mathbb{R}^2$

Let $f:\mathbb{R} \to\mathbb{R} \ be \ bounded,$ $\phi: \mathbb{R}^2 \to\mathbb{R}^2 $ be defined as $\phi(x,y)=(x,y+f(x))$ Prove that if for every bounded box $B\subset \mathbb{R}^2, \phi(B)$ ...
1
vote
1answer
39 views

Show that $\tau_c$ is a topology.

$\tau_c$ is defined as follows: $\tau_c:= \{ U\subset X \mid X\setminus U \; \text{is countable or}\; X\setminus U =X\}$ And I was wondering if this strategy works: Let $I=\{\alpha \mid X\...
3
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1answer
1k views

$[0,1)\times[0,1)$ (lower limit topology) is a regular, but not a normal topological space

Let $X=[0,1)\times[0,1)$, $\tau$ its topology with base $$\beta = \{ [a,b)\times[c,d): 0 \leq a < b \leq 1, 0 \leq c < d \leq 1 \}\;.$$ Please help me prove, that it is regular, but not a ...
6
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3answers
1k views

Regular but not normal space

This was an exam question of ours: Let $\chi$ be the set, $\chi = \left \{ a, b, c, d \right \}$. Create a topology $\tau$ on $\chi$ so that $\left ( \chi ,\tau \right )$ is regular but not ...
2
votes
1answer
396 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
3
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2answers
72 views

Product of Ultrafilters: How to show $p,q <_{RK} p\otimes q$?

Let $p,q$ be two free ultrafilter on $\mathbb{N}$, i.e. elements of $\mathbb{N}^* = \beta\mathbb{N}\setminus\mathbb{N}$. The Rudin-Keisler order is defined as follows: $p \leq_{RK} q$ iff there is a ...
5
votes
1answer
60 views
+50

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
0
votes
1answer
64 views

Existence of (non) complete metric on an interval.

I am stuck with this problem. Can anyone help me out? Thank you in advance. Question was asked in NET exam 2016 June. I am a beginner of topology. I have done b) and d). Because 0 and 1 are limit ...
0
votes
1answer
28 views

tilde mu open sets in generalized topological spaces

I found this article "on tilde mu open sets in generalized topological spaces" and from the article these sets are defined as follows. Let (X, µ) be a generalized topological space. A subset A of X is ...
1
vote
1answer
261 views

Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, prove that $Y\cup A$ and $Y\cup B$ are connected

Question is : Suppose $Y\subset X$ and $X,Y$ are connected and $A,B$ form separation for $X-Y$ then, Prove that $Y\cup A$ and $Y\cup B$ are connected. What i have tried is : Suppose $Y\cup A$ has ...
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0answers
34 views

topology of compact convergence, closed sets

Let $H(\mathbb{D})$ be the vector space of all analytic functions on the unit disk. Then the topology induced by uniform convergence on compact subsets is metrizable. Thus the following topology ...
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0answers
26 views

Dirac functional embedding

I got the following statements to show. Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with ...
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votes
0answers
51 views

$X := \prod_{i\in I} X_i \: \: $ Show that X is (path-)connected, if $X_i$ is (path-)connected $\forall i \in I$ [on hold]

Let $I$ be an indexset and $(X_i, \mathcal T_i)$ a topological Space for $i\in I$. Let $X = \prod_{i\in I} X_i$ have the product Topology. Now i have to show the following two things: $X$ is ...
1
vote
1answer
42 views

What does neighborhood/ball/closure mean in a non-metric and/or finite space?

I'm trying to understand these fundamental concepts of topology. I understand what open closed and boundary mean in a continuous space with a metric, but I'm having issues understanding the meanings ...
3
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2answers
116 views

Question on inverse limits

1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every $n\in\mathbb{N}$. Let $\...
3
votes
2answers
36 views

If $C$ is a simple closed curve lying in a simply connected open set $U$, then its interior also lies in $U$

Let $U$ be a simply connected open set in $\mathbb{R}^2$. If $C$ is a simple closed curve (a space homeomorphic to unit circle $S^1$) lying in $U$, then each bounded component of $\mathbb{R}^2\...
0
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1answer
30 views

Prove: if $X$ is a metric space, $A$ sequentially compact in $X$, then for any $a>0$ there are finite open balls of radius $a$ whose union covers $A$

Please prove: Suppose $X$ is a metric space, and $A$ is a sequentially compact set in $X$. If given a real number which is larger than zero (suppose $a>0$), there are finite open balls whose ...
1
vote
2answers
29 views

Question about proof of the tube lemma for metric spaces

Tube lemma: Let $M$ be a metric space and $K$ a compact metric space. Let $a\in M$, $a\times K\subset V\subset M\times K$, that is, suppose there is an open set $V$ between $a\times K$ and $M\times K$...
0
votes
1answer
25 views

Is the Odd-even topology weakly countably compact?

Something eludes me in the proof that the odd-even topology is weakly countable compact found here : https://proofwiki.org/wiki/Odd-Even_Topology_is_Weakly_Countably_Compact I don't understand why ...
0
votes
1answer
23 views

Perfectly Normal is hereditary

The definitions I'm working with: $(X, T )$ is called perfectly normal if whenever $C$ and $D$ are disjoint, nonempty, closed subsets of $X$, there exists a continuous function $f : X \rightarrow [...
4
votes
3answers
157 views

Extension of identity map

Suppose $z$ be the identity map on unit circle $\Bbb T$. Extending $z$ to closed unit disc $\Bbb D$. The identity map on $\Bbb D$ is an extension of $z$. Is the continuous Extension of this certain ...
0
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2answers
34 views

A theorem about one-dimensional convex sets

Suppose we have a non-empty convex set which does not consist of only one point such that it belongs to the same line, then this set is either a line segment(closed, half-open or open),a ray(closed or ...
9
votes
2answers
120 views

Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems ...
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0answers
13 views

What is the difference between Yoneda and Smyth-completeness?

What is the difference between Yoneda and Smyth-completeness. I know that Smyth-completeness is a stronger property than Yoneda-completeness; however, I was unable to find a simple definition for both ...
3
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3answers
41 views

If $(X,\tau)$ has no open connected set, must $(X,\tau)$ be totally disconnected?

Let $(X,\tau)$ a topological space and suppose that for all open sets $U \in \tau$ we have that $U$ is disconnected. Can we conclude that $(X,\tau)$ is totally disconnected, i.e. its connected ...
0
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2answers
52 views

$\phi:M\to \mathbb{R}$ continuous, $\phi(x)<\epsilon$ for $x\in X$, then $\phi(x)\le \epsilon$ for $x\in\overline{X}$

I was reading a proof that if a sequence of functions from $M$ to $N$, where $N$ is complete, converges uniformly in $X$, then they converge uniformly in $\overline{X}$, and it uses this result: $\...
2
votes
2answers
45 views

Map from 2-sphere into $(\mathbb R^3, |\cdot|)$ [duplicate]

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. Since $S^2$ is compact, $T$ ...
2
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0answers
23 views

Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$.

Describe the topology induced on the set $\mathbb N$ of positive integers by the euclidean topology on $\mathbb R$. Let $n \in \mathbb N$ then we know $(n - \frac{1}{2}, n + \frac{1}{2})$ is open in ...
0
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1answer
21 views

Find the Limit point of this exercises

Good morning, i'm working in this exercise and i solve this, but, i don't know it's fine, please how you can find the limit point? 1) $\left\{ 1-\frac{1}{n}\::\:n=1,2,3...\right\}$ Well, i say the ...
1
vote
3answers
46 views

Prove or disprove $A$ compact/closed $\implies$ $\mathcal{P}(A)$ compact/closed

For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by $$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$ Prove or disprove that $A$ ...