3
votes
1answer
42 views

Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?

Let for an open connected subset $U\subset \mathbb R^n$ and a number $\varepsilon >0$: $$ U_\varepsilon=\{x\in U: dist (x, \partial U)> \varepsilon \}. $$ Then $U_\varepsilon$ is open but in ...
0
votes
0answers
30 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
1
vote
1answer
51 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
2
votes
0answers
37 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
1
vote
1answer
40 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
0
votes
1answer
51 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
2
votes
3answers
102 views

How was born the topology from Euclidean geometry?

Good evening, I have the question arose as to create interest starting topology of Euclidean geometry, what was the interest of those who created it? Thanks for your help
4
votes
0answers
118 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
2
votes
1answer
149 views

prove that the sphere with a hair in $IR^{3}$ is not locally Euclidean at q. Hence it cannot be a topological manifold.

A fundamental theorem of topology, the theorem on invariance of dimension, states that if two nonempty open sets $U ⊂ R_{n}$ and $V ⊂ R_{m}$ are homeomorphic, then n = m. prove that the sphere with a ...
1
vote
2answers
389 views

Help understanding manifolds and topological spaces

I'm used to think about linear algebra with matrices and vectors, I don't have particular problems with geometry either, I'm having hard times understanding what is the meaning of a manifold and a ...
1
vote
1answer
62 views

Please give an example in $ \mathbb{R}^{n} $ of a set that satisfies the upper bound in the Kuratowski Closure-Complement Theorem.

Please give an example in $ \mathbb{R}^{n} $ of a set $ T $ that satisfies the upper bound of $ 14 $ in the Kuratowski Closure-Complement Theorem. Thanks!
4
votes
2answers
149 views

Congruent division of a shape in euclidean plane

Any triangle can be divided into 4 congruent shapes: http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png An equilateral triangle can be divided into 3 congruent shapes. Questions: ...
2
votes
2answers
145 views

Sets whose intersection with line segments have finite components

Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
5
votes
3answers
123 views

Special properties of $\mathbb{R}^3$

Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?
1
vote
0answers
67 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
0
votes
1answer
57 views

In an N-dimensional space filled with points, systematically find the point with highest spearmans correlation to a given-point

I asked a question exactly like this a while ago, so I do not know if it is appropriate to ask pretty much the same question with a single tweak. For the record, my first question is In an ...
2
votes
1answer
159 views

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing ...
0
votes
2answers
56 views

Applying a contraction to balls' centers increases the size of the balls' intersection?

The following statement seems clearly true, but I'm having a hard time proving it: Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix ...
1
vote
3answers
297 views

Closest Points in Euclidean Space

Let $C \subset R^n$ be the convex hull of the set of points $C' \subset R^n$ such that for every $c \in C'$, $c_i \in \{0,1\}$. Let $b \not \in C$. Is there an algorithm that will generate the ...
2
votes
2answers
360 views

High-dimensionality and intuition

Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us ...
2
votes
0answers
207 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...