17 December 2004
A Mathematical Interlude I: What is
a space?
by George E. Hrabovsky,
President MAST
Last
time we covered a lot of territory. We developed definitions
of heat and energy on the physics side, and regions, sets, and connected
sets on the mathematical side. We finished up by exploring
some of the properties of sets. As will often be the case,
we need to gain a better understanding of regions in order to understand
energy. This will require us to learn more mathematics.
Continuing the use of source [1]
from the last column, I found an article entitled, "Space" [2]. The
most useful definition explains that the term space is used to describe
any system that extends indefinitely in three directions. This
raises some more questions, but they do not seem relevant at this point;
and we need to get back towards heat. We seem to have learned
enough about space to continue, so I am making an executive decision as
project manager to go on to the next question.
There is no article on topological spaces, but the dictionary
in [1] contains the following
entry: "A set endowed with a topology." We
know what a set is, but what is a topology?
There is an article entitled "Topology" in
[3]. This article extends our notion of a space as any system
that extends indefinitely in an arbitrary number of directions. A
topological space consists of two components. The first is a set
called the underlying set [4]. The
second is a collection of subsets that satisfy three axioms. Such a collection
is called a family. The
three axioms satisfied are:
Topology Axiom 1:
The union of all subsets 
in the family 
is itself a member of the family. The subscript i
is used as a notation to count the subsets in a manner similar to a sum. So
we have a family of n subsets,
 . Let
us say that the family of subsets is the subsets of collections of points
within a circle. This axiom tells
us that the union of all possible subsets of these points are always contained
by the circle and do not include the
edge of the circle. We can write this,
Topology Axiom 2:
The intersection of any finite collection of these subsets is a member
of the family. Continuing with the example of a circle, this tells
us that so long as any overlapping collection of points is finite, those
overlaps are within the circle. We can write this,
Topology Axiom 3:
The empty set is an element of the family, and so is the underlying set. The
first is easy to understand, for a standard principle of set theory tells
us that the empty set is an element of every set. Using the example
of the circle, we see that the set of all points within the circle,  ,
forms the largest subset of the family. We can write these,
Any collection of subsets that satisfies these axioms
is called a topology.
I think that is enough to get your mind around for now. Have
a happy holiday season!
This is a wonderful little
book on topology. It is not my favorite, but it is quite good. It
begins with a very abstract presentation of set theory. While this
can be somewhat dreary, it is vitally important to understand these concepts
and, more importantly, be able to use the terminology and symbolism. Set
theory takes up the first three chapters of the book. Then we get
to the meat, the chapter on topological spaces, which is surprisingly
easy to read once you have slogged through the previous three chapters. The
remainder of the book explores different types of topological spaces and
their properties. This is a nice book, and, at less than $10, it
is a very good buy.
[1]: Mark Licker, "The Multimedia
Encyclopedia of Science and Technology," McGraw-Hill Professional
Book Group, 2000.
[2]: S. F. Singer, "Space," [1].
[3]: J. W. Morgan, "Topology," [1].
[4] Claude Berge, Topological
Spaces, Oliver and Boyd 1963 (republished by Dover Publications
in 1997.)
[5] McGraw-Hill Dictionary of Scientific and Technical
Terms, 5th Edition, in [1]. 
Created by Mathematica
(December 9, 2004) |
