Topology

Scot Free Kennedy

Main Idea

A topology on a set ${\mathcal S}$ is, in essence, a ``way to talk about subsets of ${\mathcal S}$''. That is, it is a system of subsets which is, in some sense, well-behaved: we will be able to do what we wish with the sets. Note that there are many types of things we would like to do with subsets of any particular set of interest; thus we will want to be able to talk about subsets in many different ways; thus there are many possible topologies on any one set.

Before getting in to details, let's look at a few examples:

Discrete This is simply the set of all possible subsets of ${\mathcal S}$.

Indiscrete This is the most ``coarse'' topology, consisting of $\{ {\mathcal S}, \emptyset \}$, that is, the entire set and the empty set.

Standard Toplogy on the Reals This is the familiar topology on the Real line, generated by all possible open intervals.

Lower Limit Topology on the Reals This is generated by ``half-open'' intervals, ie, sets of the form [a,b).

Basic Theory and Definitions

Connections

Literature

About this document ...

Topology

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The translation was initiated by Scot Free Kennedy on 1998-04-08