1. State whether the following statements are true or false. Justify your answers.
a) The outer measure m^ of the set is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation can also be described by an equation of the form
in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].
a) True. A singleton set (or the empty set) in (\mathbb{R}) has outer measure 0 because it can be covered by intervals of arbitrarily small total length. b) True. A finite subset of a metric space is totally bounded _________ ________ _________ ____ _____ _____ ____ _________ ____ _______ _____.
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Give an example of an algebra which is not a -algebra. Justify your choice of examples
Find in
where d is the metric given by
.
Based on the image provided, here is the transcription of the final questions:
If E is a subset of with standard metric, then show that
.
1. State whether the following statements are true or false. Justify your answers.
a) The outer measure m^ of the set is 0.
b) A finite subset of a metric space is totally bounded.
c) A connected subspace in a metric space which in not properly contained in any other connected subspace is always open.
d) The surface given by the equation can also be described by an equation of the form
in a neighbourhood of the point (0, 0).
e) A real valued function f on [a, b] is continuous if it is integrable on [a, b].
Find the interior and closure of the set of rationals in
with standard metric.
Let (X, d) be a metric space and A be a subset of X. Show that if and only if A is both open and closed.
Find the Fourier series of on
.