Math and science::Topology
Metric space. Open and closed sets
Open and closed subsets
Let
- Open in
- A subset
of is open in iff for all , there exists an such that . - Closed in
- A subset
of is closed in iff is open in .
Tom Leinster describes the openness of
Thus,is open if every point of has some elbow room—it can move a little bit in each direction without leaving .
Personally, I like the phrase: every element of an open set has a neighbourhood.
ε-balls
Open ε-balls are open, and closed ε-balls are closed. Consider trying to prove this. They are open on account of the definition of openness, not by their own definition alone, despite their names being suggestive.
Open, closed, neither or both
Subsets of a metric space can be any combination of open and closed.
Open subset, closed subset
There's no such thing as an open set that is not an open subset;
Context
Source
Tom Leinster's Topology notesComic: