do any of them please? Give an example of a function f: X rightarrow Y and a sub

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do any of them please?

Give an example of a function f: X rightarrow Y and a subset A C X such that f is open but fA, the restriction of f to A, is not open. HOMEOMORPHISMS, TOPOLOGICAL PROPERTIES Let f: X rightarrow Y and g : Y rightarrow Z be continuous. Show that if g degree f:X rightarrow Z is a homeomorphism, then g one-one (or f onto) implies that f and g are homeomorphisms. Prove that each of the following is a topological property: (i) accumulation point, (ii) interior, (iil) boundary, (iv) density, and (v) neighborhood. Prove: Let f: X rightarrow Y be a homeomorphism and let A C X have the property that Then (A subset A c X having the property is called isolated. The property of being isolated is thus a topological property.) TOPOLOGIES INDUCED BY FUNCTION Consider the following topology on Y={a,b,c,d}: T= Let X={1,2,3,4,5} f = {(1, a), (2, a), (3, b), (4,6), (5, d)}, g = {(1, e), (2, b), (3,d) (4, a), (5,c)} Find the defining subbase for the topology on X induced by f and g. Let f:X rightarrow (Y, T*). Show that if is the defining subbase for the topology T induced by the one function f, then = T. Prove: Let {fi:X rightarrowTi,)} be a collection of functions defined on an arbitraryset X . and let be a subbase for the topology TI on Yi. Then the class the following properties: (I) is a subclas of the defining subbase of the topology T on X induced by the

Explanation / Answer

the proof:(?): Suppose A is closed and x0?X?A. Take U:=X?A, an open set containing x0. NowU?A=?, so x0 is not an accumulation point. [x0 is not an accumulation point and so it is not a cluster point either.](?): Suppose A is not closed, then X?A is not open. ?x0?X?A such that no open set U?x0is contained in X?A, i.e. any open set U?x0 satisfies U?A??. So x0 is an accumulation point of A but not in A. [For this x0, note that x0?U?A because x0?A. So any open set U?x0satisfies U?A?{x0}??, i.e. x0 is a cluster point of A but not in A.]

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