Does Cantor’s intersection theorem hold for the metric space with the standard metric? Justify your answer.
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If E is a subset of with standard metric, then show that
.
For the equation , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form
.
Find the interior, closure, the set of limit points and the boundary of the set
in with the standard metric.
Find the interior and closure of the set of rationals in
with standard metric.
Let F be the function from to
defined by
Show that F is differentiable at (1, 2). Find the differential matrix of F.
Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.
If E is a measurable set and f is a simple function such that , show that