5.1: The following stat- X is disconnected.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
![Theorem 5.1: The following statements are equivalent for a topological space X:
(1) X is disconnected.
(2) X is the union of two disjoint, non-empty closed sets.
(3) X is the union of two separated sets.
(4) There is a continuous function from X onto a discrete two-point space
{a, b}.
(5) X has a proper subset A which is both open and closed.
(6) X has a proper subset A such that
ĀnX\A) = Ø.
Proof: It will be shown that (1) implies each of the other statements and that each
statement implies (1). Assume first that X is disconnected and let A, B be disjoint,
non-empty open sets whose union is X.
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FIVE / CONNECTEDNESS
((1) - (2)): B=X\A and A = X\B are disjoint, non-empty closed sets whose
union is X.
((1) - (3)): Since A and B are closed as well as open, then
ĀNB= ANB = Ø, ANB = ANB = Ø,
so X is the union of the separated sets A and B.
(1) → (4)): The function f: X → {a, b} defined by
:-
if xE A
if xEB
f(x) =
is continuous and maps X onto the discrete space {a, b}.
[(1) → (5)]: A + Ø, and
A = X\B + X
since B + Ø. Thus A is the required set. (B will do equally well.)
((1) - (6)): Either A or B can be used as the required set.
((2) - (1)): IfX = CUD where C and D are disjoint, non-empty closed
sets, then
D = X\C, C = X\D
are open as well as closed.
[(3) - (1): If X is the union of separated sets C and D, then C and D are
both non-empty, by definition. Since X = C UD and ČnD =
Ø, then ČCC, so C is closed. The same argument shows that
D is also closed, and it follows as before that C and D must be
onen as well.
[(4) → (1)): If f:X→ {a, b} is continuous, thenf~'(a) andf-'b) are disjoint
open subsets of X whose union is X. Since f is required to have
both a and b as images, both f-@) and f-'(b) are non-empty.
[(5) - (1)): Suppose X has a proper subset A which is both open and closed.
Then B = X\A is a non-empty open set disjoint from A for which
X = AUB.
(6) - (1)): Suppose X has a proper subset A for which
ĀNXA) = Ø.
Then Ā and (X\A) are disjoint, non-empty closed sets whose
union is X, and it follows as before that Ā and (X\A) are also
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