Question

In: Advanced Math

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?

Solutions

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}

Ts={ , {1}, {2}, {3},{1,2}, {1,3},{2,3}, {1,2,3}}

|Ts|=|Tx|=|X|=8.

3) | Sx| = 38= 6561

4) |Xs|= 83=512

MAXIMUM I TRIED


Related Solutions

Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set...
Calculate the relative (sub-space) topology with respect to the usual (metric) topology in R (the set of real numbers), for the following sub-sets of R: X = Z, where Z represents the set of integers Y = {0} U {1 / n | n is an integer such that n> 0} Calculate (establish who are) the closed (relative) sets for the X and Y sub-spaces defined above. Is {0} open relative to X? Is {0} open relative to Y?
Let X be a set with infinite cardinality. Define Tx = {U ⊆ X : U...
Let X be a set with infinite cardinality. Define Tx = {U ⊆ X : U = Ø or X \ U is finite}. Prove that Tx is a topology on X. (Tx is called the Cofinite Topology or Finite Complement Topology.)
Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a...
Let X be a non-degenerate ordered set with the order topology. A non-degenerate set is a set with more than one element. Show the following: (1) every open interval is open, (2) every closed interval is closed, (3) every open ray is open, and (4) every closed ray is closed. Please note: Its a topology question.
Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X),...
Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X), where P(X) := {U ⊂ X}. O is a topology on X iff O satisfies (i) X∈O and ∅∈O; (ii) ?i∈I Ui ∈ O where Ui ∈ O for all i ∈ I and I is an arbitrary index set; (iii) ?i∈J Ui ∈ O where Ui ∈ O for all i ∈ J and J is a finite index set. In a general...
X is infinite set with the finite complement topology. X is Hausdorff ??? why??? please thank...
X is infinite set with the finite complement topology. X is Hausdorff ??? why??? please thank U
Let X = NN endowed with the product topology. For x ∈ X denote x by...
Let X = NN endowed with the product topology. For x ∈ X denote x by (x1, x2, x3, . . .). (a) Decide if the function given by d : X × X → R is a metric on X where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/n where n is the least value for which xn is not equal to yn. Prove your answer. (b) Show that no...
A function f : X ------> Y between two topological spaces ( X , TX )...
A function f : X ------> Y between two topological spaces ( X , TX ) and ( Y , TY ) is called a homeomorphism it has the following properties: a) f is a bijection (one - to- one and onto ) b) f is continuous c) the inverse fucntion f  -1 is continuous ( f is open mapping) A function with these three properties is sometimes called bicontinuous . if such a function exists, we say X and Y...
Topology question: Prove that a bijection f : X → Y is a homeomorphism if and...
Topology question: Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.
26. Calculate each value requested for the following set of scores. a. ΣX X Y b....
26. Calculate each value requested for the following set of scores. a. ΣX X Y b. ΣY 1 6 c. ΣXΣY 3 0 d. ΣXY 0 –2 2 –4 27. Use summation notation to express each of the following calculations. a. Add 3 points to each score, then find the sum of the resulting values. b. Find the sum of the scores, then add 10 points to the total. c. Subtract 1 point from each score, then square each of...
Calculate the sample covariance of the following bivariate data set. X Y 27 5 15 6...
Calculate the sample covariance of the following bivariate data set. X Y 27 5 15 6 12 -2 b. Calculate the coefficient of correlation of the data set. c. Describe in words the information given to us by the sample covariance and by the sample correlation. Explain how the correlation gives us more precise information.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT