Considering the set X = {1,2,3,4,5,6,7,8}, calculate a Tx topology on X such that the following...

Considering the set X = {1,2,3,4,5,6,7,8}, calculate a Tx topology on X such that the following two conditions are met:
Tx has exactly 8 different elements, that is, | Tx | = | X | = 8
It is possible to find a subspace S of X with three elements, such that the relative topology Ts in S (with respect to Tx) has exactly 8 different elements, that is, | Ts | = | Tx | = | X | = 8

Given f: X - >> S (function of X over S), compute the smallest topology Tf over X, such that f is in C (X, S) with respect to the topology Ts over S. Is it possible to compare Tf with Tx?
Given g: S> -> X (one-to-one function of S in X), compute the smallest topology Tg over S, such that f is in C (S, X) with respect to the topology Tx over X. Is it possible to compare Tg with Ts?

Expert Solution

= {∅, {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {1,7}, {1,8}, {2,3}, {2,4}, {2,5}, {2,6}, {2,7}, {2,8}, {3,4},{3,5}, {3,6}, {3,7}, {3,8}, {4,5}, {4,6}, {4,7}, {4,8}, {5,6}, {5,7}, {5,8}, {6,7}, {6,8}, {7,8}, {1,2,3}, {1,2,4}, {1,2,5}, {1,2,6}, {1,2,7},{1,2,8}, {1,3,4}, {1,3,5}, {1,3,6}, {1,3,7}, {1,3,8}, {1,4,5}, {1,4,6}, {1,4,7}, {1,4,8}, {1,5,6}, {1,5,7}, {1,5,8}, {1,6,7}, {1,6,8}, {1,7,8},{2,3,4}, {2,3,5}, {2,3,6}, {2,3,7}, {2,3,8}, {2,4,5}, {2,4,6}, {2,4,7}, {2,4,8}, {2,5,6}, {2,5,7}, {2,5,8}, {2,6,7}, {2,6,8}, {2,7,8}, {3,4,5},{3,4,6}, {3,4,7}, {3,4,8}, {3,5,6}, {3,5,7}, {3,5,8}, {3,6,7}, {3,6,8}, {3,7,8}, {4,5,6}, {4,5,7}, {4,5,8}, {4,6,7}, {4,6,8}, {4,7,8}, {5,6,7},{5,6,8}, {5,7,8}, {6,7,8}, {1,2,3,4}, {1,2,3,5}, {1,2,3,6}, {1,2,3,7}, {1,2,3,8}, {1,2,4,5}, {1,2,4,6}, {1,2,4,7}, {1,2,4,8}, {1,2,5,6}, {1,2,5,7},{1,2,5,8}, {1,2,6,7}, {1,2,6,8}, {1,2,7,8}, {1,3,4,5}, {1,3,4,6}, {1,3,4,7}, {1,3,4,8}, {1,3,5,6}, {1,3,5,7}, {1,3,5,8}, {1,3,6,7}, {1,3,6,8},{1,3,7,8}, {1,4,5,6}, {1,4,5,7}, {1,4,5,8}, {1,4,6,7}, {1,4,6,8}, {1,4,7,8}, {1,5,6,7}, {1,5,6,8}, {1,5,7,8}, {1,6,7,8}, {2,3,4,5}, {2,3,4,6},{2,3,4,7}, {2,3,4,8}, {2,3,5,6}, {2,3,5,7}, {2,3,5,8}, {2,3,6,7}, {2,3,6,8}, {2,3,7,8}, {2,4,5,6}, {2,4,5,7}, {2,4,5,8}, {2,4,6,7}, {2,4,6,8},{2,4,7,8}, {2,5,6,7}, {2,5,6,8}, {2,5,7,8}, {2,6,7,8}, {3,4,5,6}, {3,4,5,7}, {3,4,5,8}, {3,4,6,7}, {3,4,6,8}, {3,4,7,8}, {3,5,6,7}, {3,5,6,8},{3,5,7,8}, {3,6,7,8}, {4,5,6,7}, {4,5,6,8}, {4,5,7,8}, {4,6,7,8}, {5,6,7,8}, {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,4,7}, {1,2,3,4,8}, {1,2,3,5,6},{1,2,3,5,7}, {1,2,3,5,8}, {1,2,3,6,7}, {1,2,3,6,8}, {1,2,3,7,8}, {1,2,4,5,6}, {1,2,4,5,7}, {1,2,4,5,8}, {1,2,4,6,7}, {1,2,4,6,8}, {1,2,4,7,8},{1,2,5,6,7}, {1,2,5,6,8}, {1,2,5,7,8}, {1,2,6,7,8}, {1,3,4,5,6}, {1,3,4,5,7}, {1,3,4,5,8}, {1,3,4,6,7}, {1,3,4,6,8}, {1,3,4,7,8}, {1,3,5,6,7},{1,3,5,6,8}, {1,3,5,7,8}, {1,3,6,7,8}, {1,4,5,6,7}, {1,4,5,6,8}, {1,4,5,7,8}, {1,4,6,7,8}, {1,5,6,7,8}, {2,3,4,5,6}, {2,3,4,5,7}, {2,3,4,5,8},{2,3,4,6,7}, {2,3,4,6,8}, {2,3,4,7,8}, {2,3,5,6,7}, {2,3,5,6,8}, {2,3,5,7,8}, {2,3,6,7,8}, {2,4,5,6,7}, {2,4,5,6,8}, {2,4,5,7,8}, {2,4,6,7,8},{2,5,6,7,8}, {3,4,5,6,7}, {3,4,5,6,8}, {3,4,5,7,8}, {3,4,6,7,8}, {3,5,6,7,8}, {4,5,6,7,8}, {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,5,8},{1,2,3,4,6,7}, {1,2,3,4,6,8}, {1,2,3,4,7,8}, {1,2,3,5,6,7}, {1,2,3,5,6,8}, {1,2,3,5,7,8}, {1,2,3,6,7,8}, {1,2,4,5,6,7}, {1,2,4,5,6,8},{1,2,4,5,7,8}, {1,2,4,6,7,8}, {1,2,5,6,7,8}, {1,3,4,5,6,7}, {1,3,4,5,6,8}, {1,3,4,5,7,8}, {1,3,4,6,7,8}, {1,3,5,6,7,8}, {1,4,5,6,7,8},{2,3,4,5,6,7}, {2,3,4,5,6,8}, {2,3,4,5,7,8}, {2,3,4,6,7,8}, {2,3,5,6,7,8}, {2,4,5,6,7,8}, {3,4,5,6,7,8}, {1,2,3,4,5,6,7}, {1,2,3,4,5,6,8},{1,2,3,4,5,7,8}, {1,2,3,4,6,7,8}, {1,2,3,5,6,7,8}, {1,2,4,5,6,7,8}, {1,3,4,5,6,7,8}, {2,3,4,5,6,7,8}, X}

Above show all posible topology on X.

1) ={ ∅, {1}, {2}, {3},{1,2}, {1,3},{2,3}, {1,2,3}} this form topology on X.

|| = |X| = 8.

2) S = {1,2,3}

T_s={ ∅, {1}, {2}, {3},{1,2}, {1,3},{2,3}, {1,2,3}}

|T_s|=|T_x|=|X|=8.

3) | S^x| = 3⁸= 6561

4) |X^s|= 8³=512

MAXIMUM I TRIED

Colby Messinger answered 1 year ago

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