Compact

$Cover$

{U_{λ}}_{λ \in Λ}

is cover of

X

:

X = λ \in Λ ⋃ U_{λ}

{U_{λ}}_{λ \in Λ}

is open cover of

(X, O)

:

1. U_{λ} \in O 2. X = λ \in Λ ⋃ U_{λ}

{U_{λ}}_{λ \in Λ}

is cover of

A

as subspace :

A \subset λ \in Λ ⋃ U_{λ}

{U_{λ}}_{λ \in Λ}

is open cover of

(A, O_{A})

as subspace:

1. U_{λ} \in O 2. A \subset λ \in Λ ⋃ U_{λ}

(X, O_{X})

is a compact space:

{U_{λ}}_{λ \in Λ}

is open cover of

(X, O_{X})

\Rightarrow \exists Λ_{0} \subset Λ (X = λ \in Λ_{0} ⋃ U_{λ})

A

is a compact set of

(X, O_{X})

:

{U_{λ}}_{λ \in Λ}

is an open cover of

(X, O_{X})

\Rightarrow \exists Λ_{0} \subset Λ (A \subset λ \in Λ_{0} ⋃ U_{λ})

1.2. are equivalent

1 A is compact set 2.\forall {F_{λ}}_{λ \in Λ} \subset C, \forall Λ_{0} \subset Λ (A \cap (λ \in Λ_{0} ⋂ F_{λ_{0}})) \neq = ϕ \Rightarrow A \cap (λ \in Λ ⋂ F_{λ}) \neq = ϕ

f (A)

is a compact set of

(Y, O_{Y})

A \cap F

is a compact set of

(X, O_{X})

A \times B

is a compact set of

(X \times Y, O_{X} ♮ O_{Y})