Closure, Interior and Boundary

Closure and Interior

Closedness

Let be a topological space. A subset is called closed if .

Closure

The closure of a set is the intersection of all closed sets that contain . We can say that the closure of is therefore the smallest closed set that contains .

Proposition

If is non-empty then is non-empty.

Proposition

is always closed.

Interior

The interior of , written as or , is the union of all open subsets of . It is the largest open subset of .

Proposition

.

Neighbourhood, Boundary and Limit Points

Neighbourhood

Let to be a topological space. A neighbourhood of is a set such that for some . An open neighbourhood of is an open set that contains .

Boundary

The boundary of a set is the set of all points with the property that every neighbourhood of meets both and its complement:

Theorem

Let be a topological space and . Then and .

Theorem

Let be a topological space and . Then

• A is open iff .
• A is closed iff .

Proof If is open, then every has the neighborhood that does not intersect and thus . Thus . Conversely, if , then for any we have . Thus there is an open neighborhood of not intersecting both and . Since , we have . Thus and hence is open.

Limit Point and Isolated Point

Let . A point is a limit point of if every neighbourhood of intersects . (Note that a limit point of does not need to belong to ). A point in that is not a limit point of is called an isolated point.

Theorem

Let be a topological space and . Then

Dense, Nowhere Dense and Meagre

A subset of is dense in if , is nowhere dense in if , is meagre in if it is a union of a countable number of nowhere dense sets.

e.g. is dense in (as is ). In , one-point sets are nowhere dense; so is meagre in .

Lemma

A subset of is nowhere dense if and only if is dense in .

e.g. is dense in (as is ). In , one-point sets are nowhere dense; so is meagre in . However, , so Q isn’t nowhere dense.

Convergent Sequences

Convergent Sequence

Let be a topological space. A sequence in is called convergent to if for any neighborhood of there exists an integer such that for all . We write or simply .