8. Induced topologies and factor topology

Let ( X, t ) - a topological space, Y Ì X - subset ETS in X . F Let us consider a istemu n odmnozhestv plurality of Y : t Y = ( V : V = U Į V , U Î t ).

Theorem 9. With YSTEM t Y a t opologiey n and Y .

The proof of this theorem is not difficult. T opologiya t Y called induced and whether hereditary topology and of X . P a space of ( Y , t Y ) n be ordered subspace n a space of ( X, t ).

Subsets of topological n a space of regard, to ak n ravilo, with induced T opologiey. However, it should be borne in mind that the transition to the induced topology can change the very form of open sets, their structure. So, if you take the interval [ a ; b ) with the natural topology induced from the number line, then in this topology sets of the form [ a ; c ), where a < c < b , will be open. In the original topology, they are not public.

Suppose that in a bstraktnom m nozhestve X m ezhdu N ome element s x , y Î X defined ratio x R y . E is the ratio nazyv aetsya equivalence ie If made with leduyuschie with voystva:

1 ) x R x d A l yubogo x Î X (refleksivnosg);

2 ) e If x R y , t of y R x ( symmetry);

3 ) if x R y and y R z , t on x R z (transitivity) .

The set X decomposes n and n eperesekayuschiesya plurality equiv entnyh m ezhdu with oboj e lementov, and whether to Lassen equivalence.

The set ( D x ) in Cex to Lassen e are equivalent denoted h Erez X / R .

17. Determination M nozhestvo X / R n be ordered the quotient m nozhestva X n of about elations equivalence R .

Let ( X, t ) - topological n a space of, n ust in the set X of within about elations e are equivalent R . Then n and the quotient X / R m You can to maintain natural t opologiyu following arr azom: subset V Ì ( D x ) with tinuous and s e lementov D x call on eopen t hen and only T hen, a hen of bedinenie equivalence classes D x which are put in V , to ak n odmnozhestvo X of eopen in space ( X, t ); to about Access the m nozhestvam, e stestvenno, take and n abutment m set. As is easily verified, this scoop upnost open subsets in X / the R I S THE tons opologiey and Symbol achaetsya t the R .

N reamers 23 . E If X - n Rectangle ( a ; b ) ' ( c ; d ) , and about elations equivalence R ass ano t ak h then x R y t hen and t nly t hen, to hen x and y l ezhat n and one gram of orizontal in X , m about X / R is a topological space that can be identified with the interval ( c ; d ).