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Question: i want proof number 3 as this answer b...

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Answer all items of the following problem Consider the closed unit interva [0,1] with its usual topology u, see Definition 5.4, page 33. Let J-[2, 3]. Define f I_J by f(x)-r + 2. Observe that f is a bijective function Let X = I UJ For each xe J, let B(r) = {(z)) For each re 1, let B(x)-( Wu ( f(W) {f(x)) ) : x є We u, } For example. O E 1 and 0 E 10) E ul. Since f(O ))-2, 2) and f(0) 2, Another example, consider 1. We have $ Also f14, ⅔ )) = ( 2·24 ) and f( -24 E 14 , neighborhood of . That is,佳3) U ( 21 , 2) U ( 2 , 2를 ) 93(4) 1. Prove that the family { (x) : x E X } satisfies conditions BPI. BP2, and BP2. Let T denote the unique topology on X which has the family B(x):rEX) as its neighborhood system Hint: For BP2, when you take rEUEB() arbitrary, make your classifica- tion by starting of the cases of y Observe that any element in J is an isolated point in (X, τ), hence any subset of J is open in (X, T), e., if ACJCX, then AET. Observe also that a basic open neighborhood of an element from I must contains a part which is a subset of I and a part (or more) which is a subset of J 2. Evaluate, with proof, each of the following: (a) int(27, 3] (b) (21,3] (c) int( 3) (d) (t,i 3. Prove or disprove: (X. T) is first countable 4. Prove or disprove: (X.T) is separable. Hint: See Problem 2 of 771 HW4 5. Prove or disprove: (X, T) is second countable Answer item (6) before item 7) 6. Prove that (X, T) is T 7. Prove that (X, T) is TI want proof number 3 as this answer (b) P3: Prove that the property of having an isolated point is a topological property. This is Problem Proof Let X and Y be two topological spaces such that X Y. Assume that e e X is an isolated point, e., a is open in X. Let f: XY be any homeomorphism. We show that {f(a)) is open in Y. f({e})-(f(r) :エe(a))-If(a)). Since f is a homeomorphisın. f is an open function. Thus [f(a)) is open in Y because (a) is open in the domain. P4: Consider the set of real nmbers R. Let i--T and j--2, then i R andjg R. Put X-Ruti,j. For each E P, let B(For each r E Q, let B,c):ee (a) Prove that the family { (z) :ェ X } satisfies BPI. BP2, and BP3. Let τ denote the unique topology on X which hasB):xeX) as its neighborhood system. Proof. BP1: If r X, then by the definition of B(x) BPI is satisfied. (Simple verifica BP2: Let z, U, and y be arbitrary such that r EUEB() Case 2: yEU, c) with y <cE R. Since a E [y,c), then has only two cases Either a E P or Q. If ar E P, put V- If a EQ, put V-r,c). Case 3:y-i→U-(-oo , b)U{i) with b E R. SinceエE(-00,b)U(i). then x has Subcase 3.1: . PutV U Subcase 3.2: x E P. Put V r Subcase 3.3: . Put V-, b) Case 4: y-jU-(a, oo)Ui with a e R. Since E (a, oo)Ui, then has Subcase 4.1:. Put V-U Subcase 4.2: rEP Put V rh BP3: For each r EX and each UU2 So, we put Us UinU B) we have that Uinu E B() (check) (b) Prove or disprove: ( X . T ) is first countable. The statement is true. Proof. Let E X be arbitrary. Case 1: r P. The local base from the neighborhood system zis a countable local base for (r.T) at r. Case 2: zeQ.Let B(x)={[x, z + :neN). The proof of B(r) is a countable local bao, for (X , τ ) at 2, e Q is folklore. Do it. t B(,) = {(-x,-n) U {i} : n e N). (you may choose in the right side of the left ray any qunce ()nEN so thatoo in the usual topology:.) We show that B(i) is a countable local base for (X, T) at i. Bi) is countable because N is countable. For each n N we have that (-oo.-n)Ui B(i), so (-00,-n)Ul is an open (in fact, it is a basic open) set and, by its definition, E (-o,-n)Ui for each n e N. Let V e τ be arbitrary such that i e V. Since B (i) is a local base for (X T) at i. then there exists b€ R such that (-oo, b) u {i} V. Pik an n E N such that -n<b, note that such an n exists because the set -n:nEN is unbounded below in R. Then (-oo ,-n ) u {i} E B(i) satisfies (-x.-n ) U {i} δ) U (i} I Case 4: r = j. Let BG) = { ( π+1 . oc ) U {j) : n N }. (you may choose in the left side of the right ray any sequence that ,+o in the usual topology. Note that n P for each nEN) A proof that B) is a countable local base for (X, T) at j is similar to Case 3 above but keep in your mind that the set nEN is unbounded above in R. (c) Prove or disprove: (X.T) is separable

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