Notes on Topology
Last Modified 2022-03-15
- Empty set is open:
- Whole set is open:
- Closed under arbitrary unions: If
, then .
- Closed under finite intersections: If
and is finite, then .
Definition: Topological Space, Point, Open Set
A topological space is an ordered pair
Definition: Closed Set
In other words, a set is closed if and only if its complement is open. Because taking complements is an involution (i.e.,
Note that, in topology, “open” and “closed” are not opposites! It is possible for a set to be both open and closed, and it is also possible for a set to be neither. In fact, there is a special name for sets that are both open and closed.
Definition: Clopen Set
Every topological space
Definition: Metric, Distance Function, Symmetry, Triangle Inequality
- Positive-definiteness: For all
, we have , with if and only if .
for all .
- Triangle inequality:
for all .
Definition: Metric Space, Underlying Set, Point
A metric space is an ordered pair
Definition: Open Ball