Poincare Project: Topologies and Topological Spaces
We saw in Open Sets that open subsets of a set X always follow the rules:
- the union of any collection of open sets in X is also an open set in X;
- the intersection of any finite collection of open sets in X is also an open set in X;
- the empty set is open;
- the set X itself is open.
If you pick a collection of subsets of X that follows the four rules above, that collection is said to be a topology on X. Furthermore, a set along with a choice of topology on that set is called a topological space.
The use of the word choice is an important one. A given set will (unless it is a singleton) allow multiple valid topologies. It is the choice of topology that gives a topological space its characteristics rather than the the set itself.
Consider a simply set {a, b}. The smallest possible topology would be:
{ {}, {a, b} }
In other words, the empty set and the the set itself are the only two open sets. This meets the definition of a topology and, in fact, for any set will be the smallest possible topology.
Another valid topology on {a, b} would be:
{ {}, {a}, {b}, {a, b} }
In other words, all subsets are open. This also meets the definition of a topology. For any set the topology which defines all subsets to be open will be the largest possible topology.
There are two other possible topologies that can be defined on the set {a, b}
{ {}, {a}, {a, b} }
and
{ {}, {b}, {a, b} }
Step through the four rules to convince yourself that these are valid topologies for {a, b}.
Note that, although this example has involved a small, finite set, everything here applies to infinite sets too. It is possible to define, for example, different topologies on the set of real numbers. One such topology is one that equates the open intervals with the open sets. This is by far the most intuitive topology on the reals but by no means the only one.
UPDATE: next post
Comments (2)
Marcelo on Aug. 15, 2008:
Sorry I didn't realize that this issue
had already been discussed in section
" Further Thoughts on Topologies and Open Sets"
Last Modified: Aug. 10, 2007
Author: James Tauber
Marcelo on Aug. 15, 2008:
Dear James,
first of all, I'd like to congratulate you for the nice effort you've made to work out this page in a such a wonderful pedagogical way.
My comment is about open sets. To me, it's a bit circular the way you define them. I think it's a better way to define first a topological space
associated to X as the T-collection of subsets O_i (i=1,..) of X (i.e. a class) satisfying:
1) any union of O_i's in T belongs to T
2) the intersection of a finite number of O_i (i=1,...,j) belongs to T
3) X and the empty set belong both to T
Then define the subsets O_i listed in T as OPEN sets.
That is, you don't have to mention the
word open in the 1-3 conditions above.
Moreover, you don't have to mention any ball or any metric whatsoever.
It turn's then that in R^n "open balls"
are a way to define a topology in R^n itself.