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Ninety one) a subspace. Then: (1). If {Ua { a 421}isabasis(subbasis)for. 7,{Y n U j a E forty two! } is a foundation (subbasis) for . 9}. (2) . allow A C Y. Then A is fY closed if and provided that A = Y n F, the place F is f-closed (that is, the closed units in Y are the intersections of Y with units closed in X). (3). Z, = Yn 21 ; A; = Yn A ; FrY(A) C Y m Fr(A). Yn Int(A) c hug/1),- 78 Chap. III evidence: (1) is trivial. (2). enable A be closed in Y; then A = Y Topological areas W, the place Wis open in Y there V6. 7,we ndA = Y Y, and because W: Y YO %V. Conversely, if A = Y F, F closed in X, then Y Y n %F, exhibiting is closed in Y. (3). yEA V 2 A = Y a v U(y): U(y) nA ye :25. and because A c Y, it follows that V U(y): (Y U(y)) n A seventy six 25, which exhibits that y homosexual; the consequences all opposite. the second one assertion is proved equally, and the remainder inclusions are trivial. Ex. three In Euclidean 2-space, E 2 = E1 X E1, the set E1 should be identi ed with the subset E1 x zero C E . because the topology of E2 has as foundation the open containers J x 1 , we nd from 7. 2(1) that the relative topology in E1 x zero is exactly the Euclidean topology of E1. This generalizes simply: Writing E" := E5 x E , s + t = n, the relative topology on E5 x zero C E"L is the Euclidean topology of Es X zero. we've seen in Ex. 1 that units open in a subspace don't need to be open within the complete house; the next theorem supplies asimple yet priceless case the place this can't ensue. 7. three Theorem enable Y be a subspace of X. If A C Y is closed (open) in Y, and Y is closed (open) in X, then A is closed (open) in X. evidence: For A = Y n ok, and because Y and okay are every one closed (open) in X, so is also the intersection. Ex. four A subspace Y C X is termed a discrete subspace of X each time . nine} is the discrete topology. realize that if Y is a discrete subspace of X, then Y don't need to be open or closed in X, as Y = {1/12 I n E Z+} C E1 exhibits. 7. four (Transitivity) area. A subspace of a subspace is a subspace of the complete evidence: enable Z C Y C X, and enable nine' be the topology of Z as a subspace of Y; we're to teach that . 7 = . seventy two. enable We . 7 in order that W: Zn V, the place V6. 9}. considering that V: Y U, Ue ', we nd W = Z zero YO U = Z0 U, exhibiting that W572. The speak inclusion is trivial. eight. non-stop Maps we need to relate observe that f: W(X) been contemplating topologies on one given set; we now wish assorted topological areas. Given (X, . 7X) and (Y, 9}), a map f: X > Y relates the units and likewise induces maps > W( Y), f lz . four ( Y) > of those, f"1 will be used Sec. eight non-stop Maps seventy nine to narrate the topologies, because it is the single one who preserves the Boolean operations enthusiastic about the de nition of a topology. therefore the perfect maps f: X > Y are these for which at the same time f : . seventy one, 6. 9}. officially acknowledged, eight. | Definition permit (X, 3}) and (Y, y) be areas. A map f: X > Y is named non-stop if the inverse snapshot of every set open in Yis open in X [that is, iff 1 maps :71? into EX 1 a relentless map f: X > Y is usually non-stop: The inverse picture of any set U open in Y is both :5 or X, that are open. EX. 2 enable X be any set, ninety one, .