9 . H Continuous on tobrazhenie. Homeomorphism

We discuss t EPER about EFINITIONS n Continuous display topological pros transtv. Let ( X, t ), ( Y , s ) - d va topological space with topologies t and s from ootvetstvenno. Let f : the X ® the Y - display of the plural of the EU ETS.

Definition 18 . T OVOR, h then f - continuous mapping topological n a space of, e fusion n A complete inverse image f -1 ( V ) L yubogo TCI rytogo m nozhestva V space ( Y , s ) is about Access the plural ETS space ( X, t ).

If f : X ® Y, g : Y ® Z - about tobrazheniya topological pros transtv, t of e stestvenno determined superposition g f : X ® Z n of n Ravilov ( gf ): x ® g ( f ( x )).

Theorem 10. E fusion f , g are continuous , ie about and gf continuously.

Proof l It is easy to be out of the equation

( gf ) -1 ( W ) = f -1 ( g -1 ( W )),

where W Ì Z - n roizvolnoe m nozhestvo. Let us check this equality. Let x Î ( gf ) -1 ( W ). Then g ( f ( x )) Î W Þ f ( x ) Î g -1 ( W ) Þ x Î f -1 ( g -1 ( W )). The opposite inclusion is proved similarly.

Determination 19. On tobrazhenie f : X ® Y n be ordered open (closed ), e fusion of Braz to azhdogo about Access the (closed) mnozhes Twa in X of indoor ( closed) in the Y .

Definition 20 . D wa topological n a space of, ( X, t ), ( Y , s ') , called homeomorphic , e fusion with uschestvuet mapping f : X ® Y , satisfying y words:

1 ) f : X ® Y is a bijective mapping;

2 ) f n Continuous;

3 ) f is open.

The bijectivity of the mapping f implies the existence of the inverse mapping. We denote it by g . If we take an open set U in X , then g -1 ( U ) = f ( U ) is an open set. Thus, the inverse mapping to the homeomorphic one is also continuous.

Comparison to azhdomu about Access the plurality of U space X e th of BrAZ f ( U ) n ri homeomorphism f : X ® Y establishes a bijective with Compliant m ezhdu topologies spaces X and Y . Poet y l yuboe with voystvo space X , p ormuliruemoe in m Ermin top ologii this space b udet in ernym and d A n a space of Y , homeomorphic to X , and t ak w e b udet f ormulirovatsya in terms of topology Y . T mayor of Braz, homeomorphic n a space of X and Y region adayut and dentichnymi with voystvami and with e of the m points of rhenium are n erazlichimymi.

If f : X ® Z - n Continuous about tobrazhenie topological pros transtv ( X, t ), ( Z , s ), and Y - of dprostranstvo X , m of m You can races matrivat and about tobrazhenie f : Y ® Z , to otorrhea It is called the narrowing of f on a Y and about boznachaetsya f Y .

Theorem 11 . The mapping f Y : Y ® Z is continuous.

Evidence. N ust W Î s , t hen ( f Y ) -1 ( W ) = f -1 ( W) Į W. T ak to ak f -1 ( W ) Î t , t on ( f Y ) -1 ( W ) Î t Y .

Determination 21. On tobrazhenie f : X ® Y topological pros transtv n Continuous in t ochke x 0 Î X , e fusion d To a sjakoj about the vicinity of xS O ( f ( x 0 )) t points f ( x 0 ) with uschestvuet neighborhood O ( x 0 ) t points x 0 t Which, h then f ( O ( x 0 )) Ì O ( f ( x 0 ) ).

Theorem 12 . On tobrazhenie f : X ® Y n Continuous t hen and only T hen, when on but n Continuous in a azhdoy m ochke x Î X .

Evidence. N ust f : X ® Y n Continuous, x 0 Î X - n roizvolnaya point and O ( f ( x 0 )) - n roizvolnaya about the vicinity of xS m points f ( x 0 ) . T hen n EIDET open m nozhestvo V Ì Y t Which, h the V Ì O ( f ( x 0 )) and f ( x 0 ) Ì V . N olozhim U = f -1 ( V ), U - about Access the plural ETS, x 0 Î U . T hen f ( U ) = V Ì O ( f ( x 0 )), h by Theorem 2 and proves Cont yvnost f a t ochke x 0 .

Conversely claim ust f n Continuous in a azhdoy m ochke x Î X . N ust V Ì Y - arbitrary about eopen m nozhestvo and n ust A = f -1 ( V ). T ak to ak V - Area l Juba with voey t glasses and f n Continuous in a azhdoy m ochke, t of d To every x Î A e nce of the vicinity of xS O ( x ) m points x m Which, h then f ( O ( x ) ) Ì V . Therefore, O ( x ) Ì A , that is, A is any point internal h then and d has a tkrytost A . H Continuity f d rendered.