# 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 an arbitrary set. Does there exist an extended real- valued function on P(X) that is countably additive?
Yes. The counting measure.
F : P(X) -> [0,oo]
If A c X is finite, then F(A) = the number of elements of A.
If A c X is infinite, then F(A) = oo.
Check that it is countably additive!
Let L be a countable set.
Let W = { A c L : L\A is finite }.
Describe M(W) - the monotone class generated by W.
M(W) = P(L).
page 48 in 1st measure
Let A,B be subsets of a metric space.
Suppose that A n Clo(B) is non-empty and A n B is empty.
What can we conclude from this?
A is not open
X set, let M be a countable subset of X.
W = {E c X : M\E is finite}
Describe S(W) - the s-ring generated by W.
S(W) = { E c X : MnE is countable or M\E is countable }
Let X be a set, and W c P(X), A c X.
Define "A is W-open" in the language of generalized convergence.
Let Fr(A) denote the boundary of A. (see the prev item)
A is W-open <=> Fr(A) c X\A
page 47 in gen top
Express differently:
A \ //\\t:T [ B[t] ] = ???
A \ //\\t:T [ B[t] ] = \\//t:-T [ A\B[t] ]
Express differently:
\\//t:-T [ A\B[t] ] = ???
\\//t:-T [ A\B[t] ] = A \ //\\t:T [ B[t] ]
Express differently:
( \\//t:-T B[t] ) \ A = ???
( \\//t:-T B[t] ) \ A = \\//t:-T ( B[t]\A )
Express differently:
\\//t:-T B[t]\A = ???
\\//t:-T ( B[t]\A ) = ( \\//t:-T B[t] ) \ A
Express differently:
( //\\t:-T B[t] ) \ A = ???
( //\\t:-T B[t] ) \ A = //\\t:-T ( B[t]\A )