8?. 89. Let (YER) be a subspace of (X. T}. Show that ECY is 'i'y-closed if and only if E = Y n F, where F is a Tclosed subset of X. Let (A, Ti} be a subspace of (X, 'T}. Prove that 'TA consists of the members of '1" contained in A, Lo. '1'". = {G : GCA.GE'T}. l! and only HA is a Topen subset of X. Let (r, '13,) be a subspace of {X,'!'::. For any subset A of 1',let .i and A." be the closure and interior of A 1with respect to '1'" and let (.r'liy and {Auly be the closure and interior of A with respect to Ty- Prove 5) pi}, = j. n Y, {ii} .40 = (ADJY n r0. Let A, B and C be subsets of a topological space X with C C A U B. If A, B and AUB are given the relative topologies. prove that C is open with respect to AU3 if and only if (In A is open with respect to A and Ci\": 3 is open with respect to B. EQUIVALENT DEFINITIONS OF TOPOLOGIES 91. 92. 93 . 94. 95. Prove Theorem 5.11: Let X be a non-empty set and let there be assigned to each point 136.1: a class a)!\" of subsets of X satisfying the following axioms: [All cAE. is not empty and 3) belongs to each member of My. [Ag] The intersection of any two members of My belongs to esp. [A3] Every superset of a member of as, belongs tonsil. [A4] Each member lil'EeA',JI is a superset of a member GEeA'p such that GEM\" for every 96G. Then there exists one and only one topology T on X such that cap is the T-neighborhood system of the point 306 X. vaa Theorem 5.12: Let X be a non-empty set and let k:'P{X)HP{X) satisfy the following Kuratowskl Cloaure Axioms: " [Ks] k033i} = '3. [K2] A C MA}. [\"3] klAUBl = HA} U H3}, [K4] new}: kill Then there exists one and only one topology '1" on X such that MA} will be the Tclosure of ACX. Prove: Let X be a non-empty set and let i : 'P(X) r WK] satisfy the following properties: {i} ian} = X. (if) {[411] c A, (iii) {(AUB} = 1in U :13). {iv} 1111131)): ilA} Then there exists one and only one topology T on X such that its!) will be the Tinterior of ACX. Prove: Let X be a non-empty set and let i\" be a. class of subsets of X satisfying the following properties: (i) X and Q) belong to 5". {ii} The intersection of any number of mombers of 3' belongs to :F. (iii) The union of any two members of 1? belongs to 3'". I'li'hen there exists one and only one topology 'T on X such that the members of 3? are precisely the 'Tclosed subsets of X. Let a neighborhood of a real number part be any set containing 13 and containing all the rational numbers of some open interval (a. b) where o