Reference: Chapter 1, Topology by Hocking and Young. Dover Publications, Inc. Available in Amazon India also.
Show that the mapping f used in the proof of the Theorem 1-18 is continuous.
The mapping referred to f is the following:
Consider carrying each point to
We have to prove that this map f is continuous. Clearly, the definition of continuity will work here.
Consider any point where where clearly
Let the n-tuple and
Consider an ball, viz, That is, we are considering a neighbourhood about the point x. Let the n-tuple x be given by
Then, by definition of usual metric, where is the usual norm in
So, by usual algebra, we get
. But this is an open ball of radius . So the given mapping f maps an open ball to open ball in given domain. Hence, f is open.
Given any closed interval in find a continuous mapping of into thereby proving that is connected.
We prove the classic case . Consider the set of all points x in the open unit interval is equivalent to the set of all points y on the whole real line. For example, the formula:
establishes a 1-1 correspondence between these two sets. As this map is continuous, and it maps the real line, which is connected to the above interval, the given interval is also connected.