Topological Space

Let be a set.
Topology is a collection of subsets of s.t.

• and
• 
  A topological space is the pair .
  Members of are called open sets of .
  A subset of is closed if its complement is open.

Examples

• Indiscrete topology (not induced by any metric)
• Discrete topology (induced by discrete metric)
• Cofinite topology of all sets whose complement is finite
• Cocountable topology of all sets whose complement is countable

Basic properties/defns

1. Some topologies are induced by a metric.
2. Hausdorff
3. is open if every point has an open nbd that is contained in
4. if for any nbd of we find s.t. for
5. In a Hausdorff, the limits are unique.
6. Suppose is closed and in and .
   Then no open nbd of is in
   but then is not open contradiction so it has to be .
7. HOWEVER, a set which contains all its limit points no longer has to be closed.
8. Accumulation Points
9. Interior
10. Closure
11. Subspace topology
12. Continuity in topological spaces
13. and are homeomorphic if there is a bijection between them
    such that both and are continuous
14. Product topology
15. Open map
16. Quotient space

More stuff

Connectedness
Compact
Sequentially Compact