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Let X be a set, and W c P(X), A c X.

Define "A is W-closed" in the language of generalized convergence.

Define "A is W-closed" in the language of generalized convergence.

Let Fr(A) denote the boundary of A. (see the prev item)

A is W-closed <=> Fr(A) c A

page 47 in gen top

A is W-closed <=> Fr(A) c A

page 47 in gen top

Let X be a set, W c P(X), A c X.

Prove that

A is W-open <=> X\A is W-closed

Prove that

A is W-open <=> X\A is W-closed

Use Fr(A) = Fr(X\A)

page 47 in the gen top notebook

page 47 in the gen top notebook

Let X be a set, W c P(X), A c X.

Prove that

A u Fr(A) = {x:-X : x is W-adherent to A}

Prove that

A u Fr(A) = {x:-X : x is W-adherent to A}

Notice that if x:-B, then x is W-adherent to B.

page 47 in the gen top notebook

page 47 in the gen top notebook

Let X be a set, W c P(X), A c X.

(1) A is W-closed

(2) Every point W-adherent to A belongs to A.

Prove that (1)<=>(2).

(1) A is W-closed

(2) Every point W-adherent to A belongs to A.

Prove that (1)<=>(2).

page 48 in the gen top notebook

Let X be a set, and W c P(X).

What does W* mean in the language of generalized convergence?

What does W* mean in the language of generalized convergence?

W* is the set of all W-open sets

W* c P(X)

page 48 in Gen Top

W* c P(X)

page 48 in Gen Top

Let X be a set, and W c P(X).

Prove that every set belonging to W, is W-open.

Prove that every set belonging to W, is W-open.

page 48 in gen top

Let X be a set, and W c P(X).

Let (T,>) be a directed set.

Let x:T->X be a net in X, which W-converges to k:-X.

Let p:T->T such that /\t:-T p(t)>t.

What can we say about the net {x(p(t))}t:-T ?

Let (T,>) be a directed set.

Let x:T->X be a net in X, which W-converges to k:-X.

Let p:T->T such that /\t:-T p(t)>t.

What can we say about the net {x(p(t))}t:-T ?

It also W-converges to k.

page 49 in gen top

page 49 in gen top

Let X be a set, and W c P(X).

Prove that W-convergence and W*-convergence are the same.

Prove that W-convergence and W*-convergence are the same.

page 49 in gen top

Let X be a set. Is it possible to find W c P(X) such that every
net in X W-converges to every point in X?

Yes. Let W = {X}. Or let W = {O}.

Prove that

/\x,y:-|R x*exp( x*(y^2+1)+1 ) >= -1.

/\x,y:-|R x*exp( x*(y^2+1)+1 ) >= -1.

First show that /\x:-|R x*exp(x+1) >= -1.