Closure, Interior and Boundary

Closure and Interior

Closure

Suppose is a topological space. 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. Moreover, is always closed.

Interior

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

Proposition

.

Proof Note that is open, so is closed. Therefore, to show the LHS is contained in RHS, it suffices to show . Indeed, for any , we have For the other direction, suppose is a closed set containing , then is open, so . Therefore, . In particular, .

Neighbourhood, Boundary and Limit Points

Neighbourhood

Let to be a topological space.
An open neighbourhood of is an open set that contains .
A neighbourhood of is a set containing an open neighborhood of .

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 & Isolated Point

Let . A point is a limit point of if every open neighbourhood of intersects . (Note that a limit point of does not need to belong to ). The set of all limit points is called the derived set, denoted
A point in that is not a limit point of is called an isolated point. Equivalently, A point is an isolated point of if there is an open subset of such that .

e.g. Consider , then ; For , the only limit point is .

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 if it is a union of a countable number of nowhere dense sets.

Separable Space

A topological space is separable if it has a countable dense subset.

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 isn’t nowhere dense.