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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-4
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0answers
12 views

For which points are the maps of the following metric spaces continuous? [on hold]

I saw this question and it seemed really interesting, but it had no answers: Continuity of maps between metric spaces Could you take a look?
-3
votes
0answers
26 views

prove $\bar{f^{-1}} \cdot \bar{f}=\bar{f(1)}$ [on hold]

prove $\overline{f^{-1}} \cdot \overline{f}=\overline{f(1)}$, where $f$ is a path between $a$ and $b$ in a topological space such that $f:I\rightarrow X$ and $f(0)=a,f(1)=b$ and $\bar{f}$= All paths ...
1
vote
0answers
25 views

Torus as union of circles

How do I show that the solide torus can be view as the union of circles? I know that the solid torus can be view as $D^2\times S^1$, but, how do I can show that this construction is equivalent to the ...
2
votes
2answers
66 views

Punctured plane is not simply connected

Adapt the following definition of "simply connected space" (taken from Wikipedia): A space $X$ is simply connected if it's path connected and for any continuous map $f:S^1\rightarrow X$ can be ...
0
votes
0answers
15 views

How to quantify differences/ similarities between groups of like objects?

,First let me apologize if this is not an appropriate forum for this question. Researching all the available SE sites led me here. Also, my background is not in mathematics, so layman's terms are ...
0
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1answer
25 views

Show that the open disk $D(a,r)$ is connected.

Let $a\in\mathbb{C}$ and $r>0$. Show that the open disk $D(a,r)=\{z\in\mathbb{C}\colon \vert z-a\vert<r\}$ is connected. The disk is connected if there exists a path between any two points ...
1
vote
1answer
31 views

Logical trap in R topology

We know that in metric spaces, Bolzano-Weierstrass (BW) (each infinite set owns a cluster point) and Borel-Lebesgue (BL) properties are equivalent, i.e. compactness and countably compactness are ...
1
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1answer
32 views

Show that $\mathbb{R}^3$ is a union of topological circles.

On sketch of my idea is: $\mathbb{R}^2$ fails in being such union because of the origin. Now, If I make every coordinate plane in $\mathbb{R}^3$, I mean, the planes $z = 0, x = 0, y = 0$ then I can ...
2
votes
0answers
22 views

A Homeomorphism is a Bijection between the Underlying Sets and between the Corresponding Topologies?

I haven't seen it stated as such, which is why I'm raising it here for confirmation. In most references I'm seeing a homeomorphism $\phi$ between topological spaces $(W, \mathscr S)$ and $(X, ...
0
votes
2answers
30 views

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable, where $f_x(y)$ is in fact $f(x,y)$ while $x$ is a parameter. I thought of using $\log$ in some variation, but I think it is problematic ...
2
votes
0answers
44 views

Continuity of maps between metric spaces

I want to determine for the following maps $A\rightarrow B$, for which points of $A$ the map is continuous: $(\mathbb{C},d_E)\rightarrow(\mathbb{C},d_E)$, $z\mapsto \left\{ ...
1
vote
1answer
25 views

Perfect Sets Uncountable in Metric Space

I am trying to understand the structure of the neighborhoods constructed, in Rudin's Proof. This question has been touched upon else where but it really does not go into details of the construction ...
0
votes
0answers
30 views

Why is $A=\{(x_1,x_2,…,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set?

Why is $A=\{(x_1,x_2,...,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set? This claim was shown in a solution I ran into, and I don't see how it holds. I try to follow the formal definition of nullity, ...
-4
votes
0answers
26 views

Theorem in Topology [on hold]

Please help me to prove the following (i) Interior of a set A is the set of all those points of A which are not limit points of complement of A. (ii) A point p is an exterior point of A if and ...
1
vote
1answer
23 views

Proving $[a,b]$ is closed by proving the complement is open

I'm using this definition: a subset $A$ of $\mathbb{R}$ is called open if $$ \forall x \in A, \exists \delta > 0: ] x - \delta, x + \delta [ \subset A. $$ Now, I need to prove that $[a,b]$ is ...
0
votes
1answer
22 views

cofinite topology

If we have topological space $\mathbb{R}$ equipped with the co finite topology. If we have finite subsets in consideration they are definitely closed because open sets are of the form complement of ...
-2
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0answers
17 views

Algebraic topology theorem 57.1

i read right now the theorem 57.1 on munkres book: if h:s^1 to s^1 is continuous and antipode-preserving, then h is not nulhomotopic. at the end of the proof there is written: h* is injective . so ...
2
votes
1answer
25 views

Show that $X$ is Hausdorff.

Suppose that $X$ is a space with the property that for any point $p \in X$ there is a map $f: X \rightarrow \mathbb{R}$ such that $f^{-1}(1) = \{p\}$. Show that $X$ is Hausdorff. ...
1
vote
0answers
10 views

Non-wandering point which is not in the closure of recurent points

A point $x$ in a topological dynamical system $(X,f)$ is called (positively) recurrent if $x \in \omega(x)$, where $\omega(x)$ denotes the $\omega$-limit points of $x$. $R$ denotes the set of all ...
0
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0answers
5 views

Two points in a polygonal-path-connected set can be connected with a non-intersecting polygonal path

Let $X$ be polygonal-path-connected and $x,y\in X$. So $x$ and $y$ can be connected by a polygonal path $P=\bigcup_{i=1}^n L_i$ where $L_i$ is a line segment $[x_i,x_{i+1}]$. Non-intersecting means ...
0
votes
2answers
31 views

Infinite intersection of open sets need not be open

The following is the property of an open set: The intersection of a finite number of open sets is open. Why is it a finite number? Why can't it be infinite?
0
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0answers
21 views

Definitions of proper maps

As far as I know, there several definitions of a proper map. A function $f\colon X\to Y$ is proper if it is continuous and for any space $Z$, the product $f\times \operatorname{id_Z}\colon X\times ...
1
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0answers
29 views

How can I prove two empirically derived graphs are topologically equivalent?

I have two graphs that I've derived from an empirical data set and I suspect that they're topologically equivalent. It seems much easier to show that these graphs are not equivalent than to show that ...
1
vote
2answers
63 views

Is Lebsegue Measure Translation Invariant?

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X\subseteq\mathbb{R}$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. ...
1
vote
0answers
34 views

what is relation between topology and geometry? [on hold]

what is relation between general topology and geometry ?( and example in this relation .) is there simple book in relation between general topology and geometry ?
0
votes
1answer
13 views

Properly discontinuous action on hyperbolic plane

If we have G acts properly discontinuously on hyperbolic plane $\mathbb H$, then for any point p $\in \mathbb H$, exist neighborhood V s.t. gV$\cap$V =$\emptyset$ iff gp$\neq$p. Given this, can we ...
0
votes
0answers
16 views

how to show the equivalence of density

How to show $E$ is dense if and only if $int(\mathbb{R}- E) = \emptyset$ suggestions please. I do not see how to a direct proff
2
votes
2answers
76 views

What's the mathematics behind 3D modelling? [on hold]

I'm highly interested about 3D modelling in software, and I know that it has some deep mathematics behind it too. I would like to learn what specific topics are behind it mathematically. As long as I ...
0
votes
0answers
25 views

Connected sets prove that definitions are equivalent

I found the following two definitions of connected set. I couldn't really see how they were equivalent so I tried to prove it. Definition: Two subsets $A$ and $B$ of a metric space $X$ are said to ...
1
vote
1answer
20 views

Product of regular spaces [on hold]

How to achieve demonstration of the following proposition : Every product of regular spaces is regular. Thanks for advance.
0
votes
2answers
29 views

Does a compact set with non-empty interior have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $K\subset U$ be a compact set with nonempty interior $K^{o}$, then $K$ must have a limit point in $U$. Remark: I think that the statement is true. I ...
0
votes
2answers
20 views

Does a closed set not discrete have a limit point?

My Question: Let $U\subseteq \mathbb{C}$ open and $A\subset U$ be a close set not discrete in $U$, then $A$ must have a limit point in $U$. Remark: I do not know if the statement is true. I know that ...
0
votes
1answer
58 views

Largest subset on which a function is continuous

Let $f: \mathbb{C} \to \mathbb{C}$ a function with $$f(x) =0, ~~~ \text{if} ~~ x = 0 $$ and $$f(x) = (e^x - 1)/x, ~~~\text{if} ~~x \neq 0$$ I want to determine the largest subset $A \subset ...
1
vote
3answers
35 views

Metric space $(X,d)$ with distance $D(x,S)=\inf\{d(x,y)|y\in S\}$ for $S$ subset of $X$

Let $(X,d)$ be a metric space with $S$ a non-empty subset of $X$. For $x\in X$ we define the distance $D$ between $x$ and $S$ as $D(x,S)=\inf\{d(x,y)|y\in S\}$. How do I prove that $\overline{S}$ ...
0
votes
2answers
32 views

Compact space with a discrete subspace

I'm looking for an example (or a proof of nonexistence) of a compact space with discrete and uncountable subspace.Thank you for all your answers.
0
votes
1answer
30 views

how can we define closed set or open set for a set of matrices?

Suppose we consider the set of all matrices in $M_{2}$(R) such that neither eigenvalue is real .Is the set open or closed?
2
votes
0answers
33 views

How to view Stone-Cech compactification of the real line?

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit ...
1
vote
2answers
45 views

Specific example of a space that is separable but not second countable.

A toplogical space $X$ is said to be second countable if there exists a countable basis for the topology. $X$ is separable if there exist a countable dense subset. Show that a second ...
1
vote
1answer
23 views

Converting all arcs to polygonal arcs in a plane graph

I am trying to understand a proof on the conversion of arcs to polygonal arcs in plane graphs, in the book "Graphs on Surfaces" by Mohar and Thomassen. In the book, an arc joining two points $x,y \in ...
1
vote
0answers
95 views

I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
0
votes
1answer
26 views

Proof verification: Compact set has sup and inf

I was reading this post compact set always contains its supremum and infimum There was an answer reposted as follows: As $K$ is compact, we have that $K$ is bounded. So $\sup K$ and $\inf K$ ...
0
votes
1answer
22 views

Comparing different topological spaces regarding homeomorphisms and fundamental groups.

Which of the following topological spaces are homeomorphic? Which have the same fundamental group? a) The interval $(0,1)$ and $\mathbb{E}^1$ b) The torus $\mathbb{R}^2/\mathbb{Z}^2$ ...
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1answer
13 views

Effective Topological Transformation Groups and the Group of Homeomorphisms

I'm reading Steenrod's Topology of Fibre Bundles, and on pages 6 and 7, he defines a topological group $G$ and a topological transformation group of a topological space (which I understand to be a ...
-1
votes
1answer
38 views

Show there are infinitely many distinct maximal solutions of $\frac{dx}{dt} = (3/2)x^{1/3}$ that pass through the point $(t_0,0)$

$$\frac{dx}{dt} = (3/2)x^{1/3}$$ Solve Show that given any point $(t_0,0)$ on the $t$-axis, there are infinitely many distinct maximal solutions that pass through the point. We are given: ...
1
vote
1answer
38 views

Why is the map $f(x)=e^{i2\pi x}$ from $[0, 1)$ to the unit circle continuous?

This seems to be a really silly question, I just couldn't think it straight. The definition of a continuous map: $f: X \to Y$ is continuous if for any open set $U$ in $Y$ , $f^{-1}(U)$ is open in ...
1
vote
1answer
21 views

Projection maps are open

I want to show $p_x: X\times\ Y \to X$ is an open map. Here's my proof: Let $W \subset\ X\times\ Y$ be open subset, then $W = \bigcup U_\alpha \times\ V_\beta$, for $U_\alpha, V_\beta$ are open ...
2
votes
0answers
21 views

shrinking a convex hull around a set of polygons

I'm trying to find (Or design) an algorithm that will let me, after I have a convex hull, progressively shrink the hull towards the polygon set via increasing some parameter. I.e., if we use the ...
0
votes
1answer
11 views

Suborderable space, orderable characterization proof doubt

In Orderability in the presence of local compactness, Valentin Gutev states and proves the following proposition: A suborderable space $X$ is orderable with respect to a linear order $\prec$ on it if ...
4
votes
5answers
486 views

A Compact Hausdorff Space with no Manifold Structure? [on hold]

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
0
votes
1answer
32 views

When do two subbases generate the same topology

Let $X$ be a set. If $\mathcal B_1$ and $\mathcal B_2$ are bases of subsets of $X$, it is well-known that $\mathcal B_1$ and $\mathcal B_2$ generate the same topology if and only if for any pair of ...