State whether the following statements are true or false. Justify your answers.
a) The outer measure m* of the set
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 x+y+z-sin(xyz) = 0 can also be described by an equation of the form z = f(x, y) 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].
Let's analyze each statement carefully: ### (a) The outer measure \( m^* \) of the set \( A = \{x \in \mathbb{R} : x^2 = 1\} \cup \{-3,2\} \) is 0. The set \( A \) consists of the points \( \{-1,1,-3,2\} \), which is a finite set. The Lebesgue outer measure \( m^* \) of any finite or countable set in \( \mathbb{R} \) is 0. ✅ True. ### (b) A finite subset of a metric space is totally bounded. A subset \( S \) of a metric space is totally bounded if, for every \( \epsilon > 0 \), there exists a finite cover of \( S __________ ______ ____ ______ _________ ______ _____ ______ __________ _________ _______ ___.
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