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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2
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0answers
29 views

Lebesgue covering dimension of $[0,1]$

Say, we define the Lebesgue covering dimension (LCD) like this: A set $S\in \mathbb R^n$ has LCD $d\in \mathbb N$ if and only if $d$ is the smallest natural number such that for any open cover ...
0
votes
0answers
26 views

A topology defined on collections of open covers of a topological $X$.

Is anyone familiar with a topology which is defined on collections of open covers of a topological space $(X,O)$? I am trying to define a topology induced by a linear ordering of the open covers, ...
0
votes
2answers
33 views

How Construct Clopen-Compact Bitopological Spaces?

Dear all who love general topology, In general topology we know the notion of clopen-compact spaces (introduced by A. Sostak): a topological space $(X,\tau)$ is called clopen-compact if every clopen ...
1
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2answers
74 views

Can an n dimensional object cover an n+1 dimensional object?

Is it possible for an n dimensional object to ever cover an n+1 dimensional object? For example, could a square ever cover a cube? Note: Definition of "cover" here means to completely cover the ...
0
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1answer
47 views

A question about the proof of an obvious result

This is obviously true that a local homeomorphism is a continuous map. I tried to prove it this way : Suppose $f:X \to Y$ is a local homeomorphism, then $f$ is continuous if for each $x\in X$ and ...
0
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0answers
16 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
1
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5answers
60 views

If $A$ and $B$ are compact subset of $\mathbb R$ , then so is $A+B$.

Prove the following: If $A$ and $B$ are compact subset on $\mathbb R$ , then so is $A+B:= \{a+b\mid a\in A ,b\in B\}$. I was actually thinking about first proving that if $A\subseteq \mathbb R$ is ...
1
vote
3answers
276 views

The Lebesgue Covering Lemma

Let $(X,\tau)$ a compact metric space and $\{ U_i : i \in I \}$ an open cover of $X$. Show that there is $r>0$ such that for all $a \in X$ there is an $i \in I$ such that $B_{r}(x) \subseteq ...
0
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0answers
25 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
0
votes
2answers
63 views

Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$. I already appreciate your hints/answers. Thanks
4
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3answers
386 views

Locally Compact Hausdorff Space That is Not Normal

Someone told me that locally compact Hausdorff spaces (unlike compact ones) need not be normal. Can one give me please such an example? Thank you.
0
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1answer
66 views

Question about proof on basis

I found this proof online, but I have a bit of trouble understanding it. Question: Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an ...
-1
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1answer
56 views

Topology question about open spaces of a topological space homeomorphic to the full set. [on hold]

Let $\mathcal{U}$ be an open subset of $\mathbb{R}^m$ such that there is homeomorphic $f$ from $\mathcal{U}$ to $\mathbb{R}^m$ and also $f$ is an uniformly continous function. Show that ...
1
vote
1answer
21 views

Show that a map f : (X,$\tau$) $\rightarrow$ (Y,$\tau_1$) is continuous if and only if $f^{-1}(U)\in\tau$ , for every $U\in$B1

Let (X,$\tau$) and (Y,$\tau_1$) be topological spaces and B1 a basis for the topology $\tau_ 1$. Show that a map f : (X,$\tau$) $\rightarrow$ (Y,$\tau_1$) is continuous if and only if ...
4
votes
2answers
183 views

Independence of regularity and normality in a topological space

Is it true that, in a topological space $(X, \mathcal{T})$, regularity does not imply normality and vice versa? I looked for examples to prove this; but I just don't know many examples to look ...
1
vote
1answer
181 views

Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line

Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point discrete space. Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be ...
0
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0answers
61 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
0
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0answers
27 views

Proof that a correspondence is upper hemicontinuous if and only if it's graph is closed

I'm working through a textbook (General Equilibrium Theory) where the proofing the following theorem is left as an exercise to the student - unfortunately I dont know how. Theorem 23.1: (A ...
6
votes
3answers
282 views

Given an example of a metric space in which every sphere has two centers

This is a question from Wilansky "Topology for analysis", P.15 Prob. 103 Maybe I was thinking too Euclidean, I can't come up some other "centers" of the sphere :(
3
votes
2answers
59 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
0
votes
1answer
14 views

About the interior ball condition of a convex set with C^1 boundary

Let $\Omega$ an open bounded and convex domain in $R^n$. Suppose that the boundary of this set is $C^1$. Then $\Omega$ satisfies the interior ball condition for all boundary points? Intuitively ...
2
votes
4answers
160 views

The set of points where two continuous functions agree is closed.

I want to prove that if $f,g$ are continuous functions from a topological space $(X,\tau)$ to a metric space $(Y,d)$ then the set $$ A = \{ x \in X : f(x) = g(x) \} $$ is closed. I found a very ...
0
votes
2answers
32 views

an example of a continuous bijection which is not a homeomorphism [duplicate]

I need an example of a continuous bijection $f:X \to Y$, where $X$ is NOT compact and $Y$ is Hausdorff, such that $f$ is not a homeomorphism. (It is easy to show that if $X$ is compact, then $f$ is ...
2
votes
2answers
42 views

Closed Intervals

How do topologists prove continuity of a function with the usual topology at the endpoints of a closed interval? For instance, how would a topologist prove continuity for $f(x)=x^2$ on the closed ...
2
votes
2answers
59 views

Show that two spaces are not homeomorphic

Let $H=[-1,1]\times \{0\}$ and $V=\{0\}\times [-1,0)$ in the plane. Let $T=H \cup V$. Show that $T$ is not homeomorphic to the unit interval $I=[0,1]$. My idea for this problem is that , if we remove ...
0
votes
2answers
39 views

being completely normal implies perfectly normal?

i know that being perfectly normal implies being completely normal.but is the converse true?i've read somewhere that $\overline S_\omega$ is a counterexample but i don't know how to prove it. help ...
0
votes
1answer
24 views

Fundamental group smash product

is there a way to conclude what the first fundamental group of the smash product of two path-connected spaces is? probably there should be a general way like there is for the wedge sum due to van ...
4
votes
1answer
64 views

Fundamental group of quotient of $S^1 \times [0,1]$

I have a past qual question here: Let $X = S^1 \times [0,1] /{\sim}$, where $(z,0) \sim (z^4,1)$ for $z \in S^1 = \{ z \in \mathbb{C} \colon \| z \| = 1 \}$. Compute $\pi_1(X)$. I've been trying to ...
4
votes
0answers
29 views

Properties of first-countable spaces

Hi I have questions regarding first-countable spaces. I just want to confirm something: The following are properties regarding limits and continuity of first countable spaces on Wikipedia: If $f$ ...
1
vote
1answer
32 views

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$.

If $p:E \to B$ is a covering map, and if $E$ is compact, prove that $p^{-1}(b) $ is finite for all $b \in B$. I need to verify correctness of my proof and ask if there is a more straight-forward ...
5
votes
2answers
54 views

Connectedness and compactness of a union of two sets

Let: $$A=\Big\{ (x,y) \in \mathbb R^2: 0 \le x \le 1, y=\frac{x-1}{n},\, n\in \mathbb N \Big\}$$ $$B=\Big\{ (x,y) \in \Bbb R^2: 0 \le x \le 1, y=\frac{x}{n},\, n\in \mathbb N \Big\}$$ Is $A \cup B$ ...
0
votes
1answer
44 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
1
vote
1answer
31 views

nonempty interiors can't be defined by their infinite behavior

Show that there is no topology with the property that the interior of any set is nonempty if and only if the set is infinite.
7
votes
3answers
2k views

Path connectedness and locally path connected

The Section on Covering Maps in John Lee's book "Introduction to Smooth Manifolds" starts like this: Suppose $\tilde{X}$ and $X$ are topological spaces. A map $\pi : \tilde{X} \to X$ is called a ...
0
votes
1answer
37 views

semirings and basis of a topology

Let $S$ be a semiring of subsets of a nonempty set $X$. What additional requirements must be satisfied for $S$ to be a base for a topology on $X$? Prove that if such is the case, then each member of ...
2
votes
2answers
34 views

Topology and Arithmetic Progressions

I'm self-studying from "Elementary Topology Problem Textbook" by O.Ya.Viro et al. This is Exercise 2.Lx : Consider the following property of a subset $F$ of the set $\mathbb{N}$ of positive ...
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1answer
29 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
2
votes
3answers
82 views

Symmetry of “is homotopic to” detail in the proof

Let $f,g:X\rightarrow Y$. If $f$ is homotopic to $g$ then $g$ is homotopic to $f$. Let $F:X\times I\rightarrow Y$ be a homotopy from $f$ to $g$ so $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x \in ...
1
vote
1answer
27 views

Show that composition of continuous function is continuous in product topology.

Suppose $H: X \times I \to Y$ is a continuous map of topological spaces $X,Y$ and $I = [0,1]$. And suppose $K: Y \times I \to Z$ is also a continuous map of topological spaces. I want to show that ...
0
votes
0answers
15 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
0
votes
1answer
41 views

Rudin Real and Complex Ch.2 question 16

This excerise 2.16 in Rudin is as follows: Let X be the plane with the following topology: a set is open iff it's intersection with every vertical line is an open subset of that line w/ respect to the ...
7
votes
3answers
3k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
3
votes
0answers
19 views

Ultrametric space of stochastic filtration

Let $\Omega$ be an arbitrary set and $(\mathscr F_t)_{t\in \Bbb R_+}$ be a non-decresing sequence of $\sigma$-algebras on $\Omega$ such that any subset of $\Omega$ is contained is some of them, that ...
4
votes
1answer
34 views

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$?

If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$. Is $\cup \tau_\alpha$ a topology on $X$? For all $\alpha$, $\varnothing \in \tau_\alpha$ ...
2
votes
1answer
23 views

Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?

Can someone please verify my proof? Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$? No. Let $X = \mathbb{R}$. Clearly, $\{x\} \in \tau_\infty$ ...
0
votes
1answer
29 views

Can we deduce that $X$ is $\sigma-$compact? [on hold]

Assume that a quotient space of the space $X$ is compact. Can we deduce that $X$ is $\sigma-$compact?
-2
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0answers
33 views

A question on $\sigma-$compact spaces

Let $A$ be a closed, $\sigma-$compact subspace of $X$ such that the quotient space $X/A$ is $\sigma-$compact. Can we deduce that $X$ is $\sigma-$compact?
4
votes
1answer
43 views

Homotopy classes of maps from the projective plane to $S^1 \times S^3$

I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in ...
2
votes
2answers
28 views

Prove that regular $T_1$ space is $T_2$ space.

Prove that regular $T_1$ space is $T_2$ space. Definition of $T_1$: For all $a,b\in X$, there exist $A,B\in\tau$ s,t, $a\in A, b\notin A,b\in B,a\notin B$. Definition of regular: For all $A\in ...
2
votes
3answers
556 views

Looking for examples of first countable, compact spaces which is not separable

Could someone give me some classical examples of first countable, compact spaces which is not separable? However, other examples are also welcome. Any help will be appreciated.