Cardinality - Analysis I - Lecture Notes | MATH 521, Study notes of Advanced Calculus

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Math 521: Lecture 10

Arun Ram

University of Wisconsin-Madison

480 Lincoln Drive

Madison, WI 53706

[email protected]

1 Cardinality

How big is a set?

Let S and T be sets. S and T have the same cardinality, Card(S) = Card(T ), if there is a bijective map from S to T.

Notation: Let S be a set. Then

Card(S) =







0 , if S = ∅, n, if Card(S) = Card

(

{ 1 , 2 ,... , b}

)

∞ otherwise.

Note: Even if Card(S) = ∞ and Card(T ) = ∞, one may have that Card(S) 6 = Card(T ).

A set S is finite if Card(S) 6 = ∞. A set S is infinite if S is not finite. A set S is countable if either S is finite or if Card(S) = Card(Z> 0 ). A set S is uncountable if S is not countable.

Let X be a topological space. A perfect set is a subset E of X such that

(a) E is closed, (b) if x ∈ E the x is a limit point of E.

The Cantor set is

C =

{

x ∈ [0, 1] | x 6 ∈

[ (^2) i− 1 3 k^ ,^

2 i 3 k

]

for k ∈ Z> 0 and i ∈ Z> 0 with 2i < 3 k

}

Theorem 1.1.

Card(Z> 0 ) = Card(Z≥ 0 ) = Card(Z) = Card(Q) 6 = Card(R) = Card(C).

HW: Show that subsets of countable sets are countable. HW: Show that countable unions of countable sets are countable. HW: Show that if S is countable and n ∈ Z> 0 then Sn^ is countable. HW: Show that 2Z>^0 is uncountable. HW; Show that [a, b] is uncountable. HW: Show that R is uncountable. HW: Show that the Cantor set is uncountable. HW: Show that every perfect subset of Rk^ is uncountable. HW: Show that the Cantor set is perfect.