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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0
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1answer
12 views

Show that T2-space is preserved by continuous map. [duplicate]

Let (X,$\tau$) and (Y,$\tau_1$) be topological spaces and f : (X,$\tau$)$\rightarrow$(Y,$\tau_1$) a continuous map. If f is one-to-one, prove that (Y,$\tau_1$) is Hausdorff implies that ...
0
votes
1answer
45 views

All topologies on $X=\{ a,b \}$

I am trying to find the possible topologies on $X=\{ a,b \}$. $\varnothing ,\{ a,b \}$ $\varnothing ,\{ a \},\{ a,b \}$ $\varnothing ,\{ b \},\{ a,b \}$ $\varnothing ,\{ a \},\{ b \},\{ a,b \}$ ...
0
votes
1answer
42 views

Correct proof of supremum property?

Let $u$ be an upper bound of non-empty set $A$ in $\mathbb{R}$. Prove that $u$ is the supremum of $A$ if and only if for all $\epsilon > 0$ there is an $a \in A$ such that $u-\epsilon < a$. ...
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0answers
16 views

Non-trivial compatibility which makes convex functions continuous on $\Bbb R$

Here are the definitions: Let $X$ be a set. Another set $\mathcal C\subseteq \mathcal P(X)$ is called a convexity over $X$ if $\varnothing, X\in\mathcal C$ $\mathcal C$ is closed under arbitrary ...
4
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0answers
27 views

Gluing two solid tori along their boundary resulting in a topological manifold

The following question is from a past qualifying exam. Take two solid tori $D^2 \times S^1$, and construct the space $X$ by identifying their boundaries via the map $f \colon \partial D^2 \times S^1 ...
3
votes
1answer
49 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 ...
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0answers
20 views

Family of Morse functions made constant

I'm looking for a proof of the following theorem: Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, ...
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0answers
27 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, ...
1
vote
1answer
77 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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0answers
17 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 ...
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 ...
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5answers
61 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 ...
0
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0answers
26 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 ...
1
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1answer
24 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 ...
0
votes
1answer
74 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 ...
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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 ...
0
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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
0
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0answers
35 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 proofing the following theorem is left as an exercise to the student - unfortunately I dont know how. Theorem 23.1: (A ...
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 ...
3
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2answers
63 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. ...
6
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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 :(
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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 ...
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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 ...
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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
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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 ...
2
votes
2answers
44 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 ...
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 ...
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
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0answers
30 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 $f$ ...
4
votes
1answer
65 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 ...
0
votes
1answer
44 views

Minkowski Distance Metric

Given compact sets $A$, $B$, define the Minkowski distance between the two sets as: $$ \delta(A,B):= \inf \{ r: B \subseteq \mathscr{N}_r (A) \, \, \text{and} \, \, A \subseteq \mathscr{N}_r (B) \}$$ ...
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 ...
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 ...
1
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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
1answer
38 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 ...
1
vote
1answer
30 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 ...
0
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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 ...
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
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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 ...
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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?
2
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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 ...
0
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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?
0
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1answer
37 views

Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
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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.
4
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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 ...
0
votes
1answer
10 views

Question about Neighborhood basis

In the Simon Reed text, after defining the strong operator topology it is said: "A neighborhood basis at the origin is given by the sets of the form $\{S \ | \ S \in \mathcal{L}(X,Y), ...
3
votes
2answers
41 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
2
votes
2answers
29 views

intersection number of twocompact oriented manifolds

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. ...