Question

The following are True or False statements. If True, give a simple justification. If False, justify, or better, give a counterexample.

1. (R,discrete) is a complete metric space.
2. (Q,discrete) is a compact metric space.
3. Every continuous function from R to R maps an interval to an interval.

4. The set {(x,y,z) : x² −y³ + sin(xy) < 2} is open in R³

Answer 1

TRUE .

Suppose   be a cauchy sequence in   .

In discrete space   for all   . Since    be a cauchy sequence so if we choose   then there exist   such that ,   for all but since     for all   so ,   for all .

That is the sequence   is an eventually constant sequence that is there exist   such that for all     for all .

The sequence   converges to   .

So every Cauchy sequence   in   is convergent .

Hence   is a complete metric space .

  FALSE .

In discrete metric space every every singleton set is open and so every subset in this metric space is open .

Consider the open cover of   given by ,

which do not have any finite subcover as union of any finite subcollection will of the form   which donot cover whole   as   but   .

So this cover donot have any finite subcover .

Hence , is not compact metric space .

  TRUE .

Suppose   be a continuous map and   is an interval in   .

Now if image of the interval   is not open then there exist   such that has no preimage under for some   .

Now since and   so by intermediate value property there exist such that   .

So    has preimage under , a contradiction to    has no preimage under .

Hence image of   is also an interval .

  TRUE

is defined by ,   which is a continuous map . As invese image of an open set is open under continuous map and is an open subset of   so   is open in .

Now ,   so the set   is open in   .

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