[증명] Let A and B be disjoint compact subspaces of the Hausdorff space X. There exist disjoint open sets U and V containing A and B, respectively.

명제 Proposition

Let A and B be disjoint compact subspaces of the Hausdorff space X. There exist disjoint open sets U and V containing A and B, respectively.

 

증명 Proof

For an x ∈ A,

∀ y ∈ B,  there exists neighborhoods Nx-y of x and Ny-x of y s.t. Nx-y ∩ Ny-x = ∅.

Then, { Ny-x | y ∈ B } is an open cover of B.

B is compact, so there exist a finite subcover { Ny1-x, Ny2-x, ... , Nyn-x }.

The set of corresponding x's neighborhoods is { Nx-y1, Nx-y2, ... , Nx-yn }.

 

Wx = Ny1-x ∪ Ny2-x ∪ ... ∪ Nyn-x

Zx = Nx-y1 ∩ Nx-y2 ∩ ... ∩ Nx-yn

 

Then Zx is open and a neighborhood of x and Wx ∩ Zx = ∅.

Wx is an open set containing B.

 

{ Zx | x ∈ A } is an open cover of A, and since A is compace, there exist a finite subcover:

{ Zx1, Zx2, Zx3 ... Zxn }

 

U = Zx1 ∪ Zx2 ∪ ... ∪ Zxn

V = Wz1 ∩ Wx2 ∩ ... ∩ Wxn

 

U and V are both open and U ∩ V = ∅. U contains A, and V contains B.

End of proof.

 

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