Theorem 7.33. Let X and Y be topological spaces. The product topology on X × Y is the coarsest topology on X × Y that makes the projection maps tx,Ty on X × Y continuous.
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.
How do I show 7.33? Could you explain this in great detail? Thank you!
![Definition. Suppose T and T' are two topologies on the same underlying set X. If
T C T', then we say T' is finer than T. Alternatively, we say T is coarser than J'.
We say strictly coarser or strictly finer if additionally J # T".
Definition. A function f : X
homeomorphism from X to f(X), where f(X) has the subspace topology from Y.
Y is an embedding if and only if f : X
f(X) is a
Definition. The projection maps Ax : X×Y →
by 7x(x, y) = x and ty(x, y) = y.
X and Ay : X × Y → Y are defined
Theorem 7.32. Let X and Y be topological spaces. The projection maps īx,Ty on X×Y
are continuous, surjective, and open.
In fact, the topology on the product space can be characterized as the coarsest
topology that makes the projection maps continuous.
Theorem 7.33. Let X and Y be topological spaces. The product topology on X x Y is the
coarsest topology on X × Y that makes the projection maps T,Ty on X × Y continuous.
Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X x{y}
of X × Y is homeomorphic to X.
Theorem 7.36. Let X, Y, and Z be topological spaces. A function g : Z → X × Y is
continuous if and only if nx og and ty o g are both continuous.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F7ecb8ccb-b69f-48cd-a34f-d76791f9a59e%2Fce8bazm_processed.png&w=3840&q=75)
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