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.

Define a totally bounded metric space.
For every e>0, the space can be written as a finite union of sets with diameter less than e.
page 140 in palace
What is the name for a metric space satisfying:
For every e>0, the space can be written as a finite union of sets with diameter less than e.
a totally bounded metric space
Prove that a compact metric space is totally bounded.
Suppose that the space is not totally bounded.
Construct a sequence which has no convergent subsequence.
Prove that if every sequence contained in a metric space has a Cauchy subsequence, then this space is totally bounded.
Suppose that the space is not totally bounded.
Construct a sequence which has no Cauchy subsequence.
page 141 in palace
Prove that if a metric space is totally bounded, then every sequence contained in this space has a Cauchy subsequence.
Notice that for a given e>0, every infinite set has an infinite subset with diameter less than e. Use this fact to construct a Cauchy subsequence of any sequence.
page 141 in palace
What is a finitely bound collection of sets?
Let W be a collection of sets.
W is finitely bound iff its every finite subcollection has nonempty intersection.
W is finitely bound iff every finite intersection of sets in W is nonempty.
What does it mean that a collection of sets has the finite intersection property?
Let W be a collection of sets.
(Every finite subcollection of W has non-empty intersection.)
W has the finite intersection property iff every finite intersection of sets belonging to W has nonempty intersection.
(1) W is a collection of sets with the finite interec
Prove that in a sequentially compact metric space every finitely bound collection of closed sets has nonempty intersection.
page 142 in golden gate
Let W be a finitely bound collection of closed subsets of a sequentially compact metric space X. We know that the space is totally bounded. Hence, for every natural n, X is a finite union of balls of radius 1/n. Suppose that for each of these balls there is a set in W which has empty intersection with this ball. By the nature of W, this finite collection of
If there is a point belonging to the intersection of a collection of sets, we say that this collection has ___________.
it has nonempty intersection
When no point belongs to the intersection of a collection of sets, we say that the collection has ___________.
it has empty intersection