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 = [0,1). Define d(x,y) = |x/(1-x) - y/(1-y)|.
Show that (X,d) is a metric space.
f : [0,1) -> [0,oo)
f(x) = x / (1-x)
Show that f is 1-1.
Conclude that d is a metric.
Let X = [0,1). Define d(x,y) = |x/(1-x) - y/(1-y)|. (X,d) is a metric space.
Is (X,d) complete?
Yes.
f : [0,1) -> [0,oo)
f(x) = x / (1-x)
Use the fact that f is a bijection.
Consider an infinitely countable complete metric space.
How many isolated points does it have?
Infinitely many.
If you remove an isolated point from a complete metric space,
then the remaining space is still complete.
Use the Baire Category Theorem to obtain one isolated point.
Consider a complete metric space which has no isolated points.
Can it be countable?
NO.
A countable complete metric space must have an isolated point.
(see the previous item)
Let A be a closed subset of |R. Suppose that it is infinite and has no isolated points. Can A be countable?
NO.
A closed subset of |R is complete.
A countable complete set must have an isolated point.
(see the previous item)
Let (X,d) be a metric space. A c X.
What does it mean that A is nowhere-dense?
Int(Clo(A)) = O
(the interior of the closure of A is empty)
The closure of A contains no ball.
nowhere-dense = rare
What do you call a subset of a metric space whose closure contains no ball?
1) nowhere-dense
2) rare
Let (X,d) be a metric space. A c X.
What does it mean that A is meager?
A meager set is a countable union of rare sets.
(The closure of a rare set contains no ball.)
(Every rare set is meager.)
Let (X,d) be a metric space. A c X.
What does it mean that A is rare?
Int(Clo(A)) = O
(the interior of the closure of A is empty)
The closure of A contains no ball.
rare = nowhere-dense
What do you call a subset of a metric space that is a countable union of rare sets? (A rare set is one whose closure contains no ball.)
1) meager
2) of the first category