2. Interior, Closure, Boundary; Neighbourhoods, Density; Completeness

Front 1

Definition interior, closure

Back 1

Let Y ⊂ (X, τ ). Then the interior Int(Y) and closure Cl(Y) of Y are defined by:

Int(Y) = ∪{U | U open, U ⊂ Y }

Cl(Y) =\{C | C closed, C ⊃ Y }.

The points of int(Y) are called interior points of Y, those of Cl(Y) adherent points of Y .

Front 2

Lemma interior and closure under complements

Back 2

X\Int(Y) = Cl(X\Y)

X\Cl(Y) = Int(X\Y)

Front 3

Definition boundary

Back 3

If Y ⊂ (X, τ ) then the boundary ∂Y of Y is∂Y = Cl(Y)\Int(Y).

Front 4

Basic properties boundary

Back 4

Let Y ⊂ (X, τ ). Then:

(i) ∂Y = Cl(Y) ∩ (X\Int(Y) ) = Cl(Y) ∩ Cl(X\Y) = ∂(X\Y).Thus a subset and its complement have the same boundary.

(ii) ∂Y is closed.

(iii) Y = Y ∪ ∂Y and Int(Y) = Y\∂Y.

(iv) ∂Y = ∅ ⇔ Cl(Y) = Int(Y) ⇔ Y is clopen.

(v) ∂Y = X ⇔ Cl(Y) = X and Int(Y) = ∅.

Front 5

Basic properties closure map

Back 5

The map cl: Y --> Cl(Y) (defined on P(X)), has the following properties:

(i)cl(∅) = ∅.

(ii) cl(Y) ⊃ Y for every Y ⊂ X.

(iii) cl(cl(Y)) = cl(Y) for every Y ⊂ X.

(iv) cl(A ∪ B) = cl(A) ∪ cl(B).

(v) cl(A ∩ B) ⊂ cl(A) ∩ cl(B).

Front 6

Definition (open) neighborhood

Back 6

Let x ∈ (X, τ ).

(i) An open neighborhood of x is a U ∈ τ such that x ∈ U. Theset of open neighborhoods of x is denoted Ux.

(ii) A neighborhood of x is a set N ⊂ X that contains an open neighborhood of x. The set of allneighborhoods of x is denoted Nx.

Front 7

Lemma closure and neighborhoods

Back 7

Let Y ⊂ (X, τ ). Then x ∈ cl(Y) if and only if N ∩ Y =/= ∅ for every (open) neighborhoodN of x

Front 8

Lemma empty intersections with an open set

Back 8

If U ∩ V = ∅ and U is open then U ∩ cl(V) = ∅.

Front 9

Definition dense

Back 9

A set Y ⊂ (X, τ ) is called dense (in X) if cl(Y) = X.

Front 10

Characterisation denseness using open sets

Back 10

Y ⊂ (X, τ ) is dense if and only if Y ∩ W=/= ∅ whenever ∅ =/= W ∈ τ

Front 11

Lemma's denseness and open sets and corollary

Back 11

(i) If Y ⊂ X is dense and V ⊂ X is open then V ⊂ cl(V ∩ Y).

(ii) If V, Y ⊂ X are both dense and V is open then V ∩ Y is dense.

COROLLARY:

Any finite intersection of dense open sets is dense

Front 12

Definition accumulation points and derived set

Back 12

If(X, τ)is a topological space and Y ⊂ X, a point x ∈ X is called an accumulation pointof Y if every neighborhood of x contains a y∈ Y, y=/=x.

The set of accumulation pointsof Y is called the derived set Y0.

Front 13

Definition irreducible space

Back 13

Let (X, τ) be a topological space. The following equivalent statements define an irreducible space:

(i) If C, D ⊂ X are closed and X = C ∪ D then C = X or D = X.

(ii) If U, V ⊂ X are non-empty open sets then U ∩ V =/= ∅.

(iii) Every non-empty open U ⊂ X is dense.

If (X, τ) is not irreducible, it is reducibel, and there are disjoint U, V ⊂ X non-empty and open.

Front 14

Definition T0-space

Back 14

Let (X, τ) be a topological space. The following equivalent statements define a T0-space:

(i) Given x, y ∈ X, x =/= y, there is a U ∈ τ containing precisely one of the two points. (I.e. allpoints are distinguished by τ .)

(ii) If x =/= y then {x} =/= {y}.

Front 15

Cantor's intersection theorem (a topological characterisation of completeness)

Back 15

Let (X, d) be a complete metric space. Let {Cn} be a sequence non-empty, closed subsets of X, with: if n ≥ m => Cn ⊂ Cm. Also, let diam(Cn)-->0. (diam(Cm)=sup(d(x,y)) for x,y in Cn). Then:

∩n(Cn)={x} for certain x in X.