# Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Let X be a set, and W c P(X), A c X.
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
Let X be a set, W c P(X), A c X.
Prove that
A is W-open <=> X\A is W-closed
Use Fr(A) = Fr(X\A)
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}
Notice that if x:-B, then x is W-adherent to B.
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).
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?
W* is the set of all W-open sets
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.
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 ?
It also W-converges to k.
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.
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.
First show that /\x:-|R x*exp(x+1) >= -1.