7 January 2005

Mathematical Interlude II: What is closure?

George E. Hrabovsky, President MAST

Where We Have Been

Last time I explained spaces and topological spaces. We need to cover some more math before we can return to physics.

Session 3: What is closure?

To begin, any set of points that does not contain the boundary of the set is called an open set. Any set that contains its boundary is a closed set. Last time we took a circle as an example. The set of points contained within a circle is an open set; the set of bounds bounded by a circle, and including the circle itself, is a closed set. Using the reference [1] below, we can see that a closure is the intersection of all subsets of the underlying set of a topological space such that each subset of the intersection is closed and the closure is itself a subset of the underlying set. We can write this symbolically as,

where is the topological space, is a subset of the topological space, and is a subset in the intersection.

What is euclidean space?

First, we define an n-tuple as a sequence of n numbers, each number representing a value along a coordinate axis. A position vector describing the position of a particle moving through a space described by cartesian coordinates is an example of a 3-tuple.

Using source [3] we can define an n-dimensional euclidean space as the set of all real n-tuples.

What is a region?

We can now return to one of the fundamental aspects of the definition of heat. That is, "What is a region?" Again, the answer is, "A nonempty, open, connected set in euclidean space."  Though we do not know the subtleties of this definition, we do have enough to understand that this means that we have a set of points in space, not including their boundary, and including all elements within the space other than its boundary; and that these points can be described using the language of vectors.

29. What is vector?

We are now better equipped to deal with the physics next time.

Book Review: General Topology

Stephen Willard, General Topology, Addison-Wesley (1970, renewed by Stephen Willard in 1998 and reprinted by Dover Publications in 2004).

In my opinion this is the best book on general topology. The book begins with a nice introduction to set theory and metric spaces. It then devotes two chapters to topological spaces and their properties. These are followed by chapters on convergence, separation and countability of sets, and compactness. Then there is a chapter on metrizable spaces (this is a property that applies distance rules to general spaces). This is followed by connectedness (a property that every two points in a space are connected by a completely smooth function). The next chapter is on uniform spaces (that is, spaces that admit continuous functions uniformly throughout the space). The final chapter is on function spaces (these are spaces whose points are all functions that smoothly change with respect to some parameter).

References

[1] Stephen Willard, General Topology, Addison-Wesley (1970, renewed by Stephen Willard in 1998 and reprinted by Dover Publications in 2004).

[2] McGraw-Hill Dictionary of Scientific and Technical Terms, 5th Edition, in Mark Licker, The Multimedia Encyclopedia of Science and Technology, McGraw-Hill Professional Book Group (2000).

[3] Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics CRC Press (1999).

Created by Mathematica  (December 28, 2004)