Given X = a, b, c and τ = ∅, a, b, c, a, b, a, c, b, c, a, b, c, determine if τ is a topology on X.

Here's the solution for each problem. Problem 1: Given X = \a, b, c\ and = \, \a\, \b\, \c\, \a, b\, \a, c\, \b, c\, \a, b, c\\. We need to determine if is a topology on X by checking the three axioms of a topology. Step 1: Check if and X . We see that and X = \a, b, c\ . Axiom 1 is satisfied. Step 2: Check if the intersection of any two sets in is also in . Let A, B . We need to verify if A B . Since contains all possible subsets of X, the intersection of any two subsets of X will also be a subset of X. Therefore, their intersection will be in . For example: \a, b\ \b, c\ = \b\ \a\ \c\ = Axiom 2 is satisfied. Step 3: Check if the union of any collection of sets in is also in . Let \A_i\_i I be any collection of sets in . We need to verify if _i I A_i . Since contains all possible subsets of X, the union of any collection of subsets of X will also be a subset of X. Therefore, their union will be in . For example: \a\ \b\ = \a, b\ \a, b\ \c\ = \a, b, c\ = X Axiom 3 is satisfied. Since all three axioms are satisfied, is a topology on X. Problem 2: Given X = \a, b, c, d\ and = \, X, \a, b\, \a, c\, \a, b, c\\. We need to determine if is a topology on X by checking the three axioms of a topology. Step 1: Check if and X . We see that and X = \a, b, c, d\ . Axiom 1 is satisfied. Step 2: Check if the intersection of any two sets in is also in . Let's consider the intersection of \a, b\ and \a, c\ . \a, b\ \a, c\ = \a\ However, \a\ is not an element of . Therefore, Axiom 2 is not satisfied. Since Axiom 2 is not satisfied, is not a topology on X. Final Answer: 1. is a topology on X. 2. is not a topology on X.