Reference: Chapter 1, Topology by Hocking and Young. Dover Publications, Inc. Available in Amazon India also.

Question 1:

Show that the mapping f used in the proof of the Theorem 1-18 is continuous.

Proof 1:

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.

Question 2:

Given any closed interval in find a continuous mapping of into thereby proving that is connected.

Solution/Proof:

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.

Cheers,

Nalin Pithwa

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