Final answer:
The collection S of sets X\{x}, where x is an element of X, is a subbasis for the finite-closed topology since any finite set in X can be represented as an intersection of elements from S and any open set in the finite-closed topology can be formed using unions and intersections of elements from S.
Step-by-step explanation:
The student's question involves proving that the collection S of all sets derived from the original set X by removing a single element (X\{x}) is a subbasis for the finite-closed topology on X. To prove that S is a subbasis for the finite-closed topology, we need to show that the arbitrary unions and finite intersections of elements from S generate the finite-closed topology on X. In other words, we should demonstrate that any set closed under finite intersections can be represented as an arbitrary union of elements from S.
Proof that S is a Subbasis for the Finite-Closed Topology on X
Step 1: Any finite set F in X can be represented as the intersection of the sets in S containing the complement of F. That is, F can be written as X\{f_1} ∩ X\{f_2} ∩ ... ∩ X\{f_k} for all f in F.
Step 2: Since finite intersections of sets in S are in the topology generated by S, every finite set is in the finite-closed topology, and hence, S is a subbasis for the finite-closed topology.
Step 3: Every complement of a finite set is an open set in the finite-closed topology, which can be written as the union of elements in S. Thus, every open set in the finite-closed topology can be formed by arbitrary unions and finite intersections of elements of S.
From these steps, we conclude that S is indeed a subbasis for the finite-closed topology on X, since the open sets in this topology are exactly the complements of the finite sets in X, and these complements can be constructed using arbitrary unions and finite intersections of elements from S.