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Topology Terminology

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,102
The Astral plane
I don't often cross post but here we go!

This is should be a simple one. I know what I'm looking for but I don't know what to call it.

I have a member on PHF that I am talking with and I'd like to know the terminology used in the next two definitions.

1) I want to compactify the real numbers by defining \(\displaystyle - \infty\) and \(\displaystyle \infty\) to belong to the compactified set such that, for any member "a" in the compactified set we have that \(\displaystyle -\infty \leq a \leq \infty\). My model is the "hyperintegers" \(\displaystyle [ -\infty, \text{ ... } -1, 0, 1, \text{ ... } , \infty ] \). I know that the hyperreals also contain infinitesimals but I don't actually need them for the discussion.

2) Can I say that the reals and the compactified real numbers defined above are "locally homeomorphic?"

I don't need to get too deep into either concept, I just need the correct terminology so I can talk about it unambiguously.

Thanks!

-Dan
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,653
Leeds, UK
I'd like to know the terminology used in the next two definitions.

1) I want to compactify the real numbers by defining \(\displaystyle - \infty\) and \(\displaystyle \infty\) to belong to the compactified set such that, for any member "a" in the compactified set we have that \(\displaystyle -\infty \leq a \leq \infty\).
I would call that set the "2-point compactification" of the real numbers. If you use a map like $x\mapsto \tan\frac{\pi x}2$ to provide a homeomorphism between the open unit interval $(0,1)$ and the real line $\Bbb R$, then that extends to a homeomorphism between the closed unit interval $[0,1]$ and the 2-point compactification of $\Bbb R$.

2) Can I say that the reals and the compactified real numbers defined above are "locally homeomorphic?"
That certainly holds in a neighbourhood of any point of $\Bbb R$. It obviously doesn't make sense at the points $\pm\infty$ of the 2-point compactification.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,632
Leiden
I don't often cross post but here we go!

This is should be a simple one. I know what I'm looking for but I don't know what to call it.

I have a member on PHF that I am talking with and I'd like to know the terminology used in the next two definitions.

1) I want to compactify the real numbers by defining \(\displaystyle - \infty\) and \(\displaystyle \infty\) to belong to the compactified set such that, for any member "a" in the compactified set we have that \(\displaystyle -\infty \leq a \leq \infty\). My model is the "hyperintegers" \(\displaystyle [ -\infty, \text{ ... } -1, 0, 1, \text{ ... } , \infty ] \). I know that the hyperreals also contain infinitesimals but I don't actually need them for the discussion.

2) Can I say that the reals and the compactified real numbers defined above are "locally homeomorphic?"

I don't need to get too deep into either concept, I just need the correct terminology so I can talk about it unambiguously.

Thanks!

-Dan
This is not really my area of expertise, but let me add this:
The integers do not really form an 'easy' topological space.
That is, homeomorphisms do not apply until we get an actual topological space into place.
If we do limit ourselves to integers, we need to adopt for instance the so called discrete topology.
It means that any set that contains an integer is both open and closed.
Moreover, we cannot 'compactify' that set.
That is because one of the properties of the discrete topology is:
A discrete space is compact if and only if it is finite.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,102
The Astral plane
This is not really my area of expertise, but let me add this:
The integers do not really form an 'easy' topological space.
That is, homeomorphisms do not apply until we get an actual topological space into place.
If we do limit ourselves to integers, we need to adopt for instance the so called discrete topology.
It means that any set that contains an integer is both open and closed.
Moreover, we cannot 'compactify' that set.
That is because one of the properties of the discrete topology is:
A discrete space is compact if and only if it is finite.
Yes, this is true. My comment about the supposed "hyperintegers" is not meant to be a real thing. I was hoping it would illuminate the set that I was trying to talk about in case there was confusion.

I have a member on PHF that is apparently trying to use the hyperreals but the reference he posted was in Chinese and I couldn't think of a way to get it over to Google translate. The clue to the hyperreals was an equation: \(\displaystyle \dfrac{1}{\infty} = 0\). I sort of went on to the rest of it myself and had to check the terminology for "local homeomorphism" which I couldn't find a source for. (My web searching skills are about on the same miserable level as my ability to do experimental Physics.)

Thanks for the comment!

-Dan
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,653
Leeds, UK
If we do limit ourselves to integers, we need to adopt for instance the so called discrete topology.
It means that any set that contains an integer is both open and closed.
Moreover, we cannot 'compactify' that set.
That is because one of the properties of the discrete topology is:
A discrete space is compact if and only if it is finite.
It's true that the space $\Bbb Z$ of integers cannot have a compact topology. But as a topological space with the discrete topology it does have many compactifications. A compactification is a compact space that contains the given space as a dense subspace.

The "smallest" compactification of $\Bbb Z$ is the 1-point compactification, obtained by adding a single point $\infty$ whose open neighbourhoods are the complements of the finite subsets of $\Bbb Z$. At the opposite extreme is the Stone–Čech compactification, which is an enormous space (in the sense that it has a huge cardinality) but still has $\Bbb Z$ as a dense subset. In between there is a whole hierarchy of compactifications, including for example the Bohr compactification, which retains and extends the additive group structure of $\Bbb Z$.