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 W c P(X). Let M c P(X) be a collection of W-open sets.

What can we conclude?

What can we conclude?

\\//M is W-open

page 56 in gen top

page 56 in gen top

Let X c P(X).

Which of these statements are true?

(1) the empty set is W-closed

(2) the empty set is W-open

(3) X is W-closed

(4) X is W-open

Which of these statements are true?

(1) the empty set is W-closed

(2) the empty set is W-open

(3) X is W-closed

(4) X is W-open

all are true

Let W c P(X) and M c P(Y). Let f : X -> Y.

(1) /\a:-X [ if net x in X W-converges to a, then net f(x) M- converges to f(a) ]

(2) /\BcY [ B is M-closed ==> f-1(B) is W-closed ]

Prove that (1)=>(2).

(1) /\a:-X [ if net x in X W-converges to a, then net f(x) M- converges to f(a) ]

(2) /\BcY [ B is M-closed ==> f-1(B) is W-closed ]

Prove that (1)=>(2).

page 57 in gen top

Let W c P(X) and M c P(Y). Let f : X -> Y.

(1) /\BcY [ B is M-open ==> f-1(B) is W-open ]

(2) /\a:-X [ if net x in X W-converges to a, then net f(x) M- converges to f(a) ]

Prove that (1)=>(2).

(1) /\BcY [ B is M-open ==> f-1(B) is W-open ]

(2) /\a:-X [ if net x in X W-converges to a, then net f(x) M- converges to f(a) ]

Prove that (1)=>(2).

page 57 in gen top

Let W c P(X) and M c P(Y). Let f : X -> Y.

(1) /\BcY [ B is M-closed ==> f-1(B) is W-closed ]

(2) /\BcY [ B is M-open ==> f-1(B) is W-open ]

Prove that (1)=>(2).

(1) /\BcY [ B is M-closed ==> f-1(B) is W-closed ]

(2) /\BcY [ B is M-open ==> f-1(B) is W-open ]

Prove that (1)=>(2).

page 57 in gen top

Let W c P(X), A c X, x:-X.

(1) x is W-adherent to A

(2) /\FcX [ F is W-closed and A c F ==> x:-F ]

Prove that (1)=>(2).

(1) x is W-adherent to A

(2) /\FcX [ F is W-closed and A c F ==> x:-F ]

Prove that (1)=>(2).

page 58 in gen top

Let W c P(X), A c X, x:-X.

(1) /\FcX [ F is W-closed and A c F ==> x:-F ]

(2) /\GcX [ G is W-open and x:-G ==> AnG != O ]

Prove that (1)=>(2).

(1) /\FcX [ F is W-closed and A c F ==> x:-F ]

(2) /\GcX [ G is W-open and x:-G ==> AnG != O ]

Prove that (1)=>(2).

page 58 in gen top

Let W c P(X), A c X, x:-X.

(1) /\GcX [ G is W-open and x:-G ==> AnG != O ]

(2) x is W-adherent to A

Prove that (1)=>(2).

(1) /\GcX [ G is W-open and x:-G ==> AnG != O ]

(2) x is W-adherent to A

Prove that (1)=>(2).

page 58 in gen top

Let W c P(X), a:-X.

Suppose /\B,A:-W [ B n A :- W ].

Let K = {A:-W : a:-A} be non-empty.

Let f : K -> X satisfy /\B:-K f(B):-B.

What interesting fact can we conclude from this setup?

Suppose /\B,A:-W [ B n A :- W ].

Let K = {A:-W : a:-A} be non-empty.

Let f : K -> X satisfy /\B:-K f(B):-B.

What interesting fact can we conclude from this setup?

1) (K,c) is a directed set, where c is the inclusion relation

2) the net f W-converges to a

page 60 in gen top

2) the net f W-converges to a

page 60 in gen top

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

What does it mean that W is a base for G ?

What does it mean that W is a base for G ?

/\E:-G \/BcW E = \\//B

Every set from G can be written as a union of sets from W.

(a union of any cardinality, that is)

page 61 in gen top

Every set from G can be written as a union of sets from W.

(a union of any cardinality, that is)

page 61 in gen top