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 ______ _________ ___ ______ ______ ______ _________ __________ _______ ____ ________ _______.
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Give an example of a family fi of subsets of a set X which has finite intersection property. Justify your choice of example.
Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.
Obtain the second Taylor’s series expansion for the function given by
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.
Let F be the function from to
defined by
Show that F is differentiable at (1, 2). Find the differential matrix of F.
Which of the following subsets of are compact w.r.t. the metric given against them. Justify your answer.
i) in
of
with standard metric.
ii) with discrete metric.
iii) with standard metric.
Find in
where d is the metric given by
.
Based on the image provided, here is the transcription of the final questions: