A Blast from the Past, continued

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This is the second topological post migrated here from The Auld Blogge on LiveJournal, now virtually inaccessible to me. (And one presumes to others.) We shall see if the illustrative diagrams survive the journey.

Dec. 6th, 2009 at 9:43 PM

Putting on My Top Hat

I warned you. Nostalgia is a terrible force. Could we but harness it, our energy problems would be solved. Ah, the good old days...

Topologies

Topology is a branch of mathematics concerned with whether or not two points are close together, and how this closeness is affected by mappings from one space to another. On what basis can we say that two point a and b are "close"?

A topology on X is a family of subsets that constitute the "open" subsets of X. The familiar open sets of the real number line (a,b) constitute such a topology on E^1. The difference between the set of all real numbers and Euclidian 1-space is the open set structure, or topology. This is the difference between a set and a space.

Now it's nice to know that the family of all open sets consists of open sets. What topology does is build this up from beneath. IOW, we start with no preconception of what constitutes an "open" set and a set of rules for what a topology is. Basically, a topology is a family of sets that is closed under union (each union of members of the family is a member of the family) and closed under finite intersection (each finite intersection etc.) And, oh yeah, Ø and X are members, too.  It turns out (fortunately) that open sets are open sets under these rules. Phew.

A set can have more than one topology. The coarsest (smallest) topology is just that consisting of {Ø,X}, that is: the null set and the space itself. In such a topology, there are no small neighborhoods. If you want to say where a point is, it's in X. Sorry, can't pin it down any closer. This is called the indiscrete topology. The finest (largest) topology is P(X), the Power Set, which consists of the set of all subsets of X. This is the discrete topology.

Consider the set X={0,1} consisting of exactly two points with the discrete topology. 

T = P(X) = {Ø, 0, 1, X} This space is called 2. 

(Nice to know where 2 comes from when you start out with only 0 and 1.....)
 

The same set with the topology T = {Ø, 0, X} is called Sierpinski Space or S. Notice that {1} has no small neighborhoods in Sierpinski space, since the only open set containing {1} is X itself.

That's enough of that.

Functions

A function is a continuous map from one topological space to another. Notation is f:Y→Z. For example, the square maps E^1 into E^1 (actually into the non-negative part of E^1) by mapping 0 to 0, 1 to 1, 2 to 4 and so on and in between. This is illustrated below for a few points in Y.

This illo shows how a few open sets in Y are mapped into open sets in Z. For example:
f:(-2.8284271etc, +2.8284271etc) → (0,8)
f:(1,2) → (1,4)
f:(-2,-1) → (1,4)
f:(-1,+1) → (0,1)

Lastly, consider other functions mapping the same interval (0, 1.5 etc.) into Z.
-4 maps it into (-4) a "constant" map
Y^2 maps it into (0, 2.25)
-Y maps it into (-1.5, 0)
Y+2 maps it into (2,3.5)4Y maps it into (0,6)


Function Spaces

It is natural to ask: what sort of topological space can we make of the set of all continuous functions from Y->Z.
In what sense can we say two functions f and g are "near" each other?
Who cares?

Tune in again next time for the next exciting installment in this mesmerizing topic. 

Alas

No third installment was posted. It began to seem all too esoteric a topic of an internet post.