Share
Explore BrainMass

# Prove that union I: = Uα Iα is an open interval.

Fix a point p in R. Let { I&#945; } be a ( possibly infinite ) collection of open intervals I&#945; = ( c&#945; , d&#945; )
which is a subset of R, such that p&#1028; I&#945; for all &#945;.

Prove that the union I: = U&#945; I&#945; is also an open interval ( possibly infinite ).

Hint: Consider c: = inf&#945; c&#945; and d: = sup&#945; d&#945; and show that I = ( c, d ).

#### Solution Summary

This solution is comprised of a detailed explanation of the union of open intervals is also an open interval.
It contains step-by-step explanation for the following problem:

Fix a point p in R. Let { I&#945; } be a ( possibly infinite ) collection of open intervals I&#945; = ( c&#945; , d&#945; )
which is a subset of R,such that p&#1028; I&#945; for all &#945;.
Prove that the union I: = U&#945; I&#945; is also an open interval ( possibly infinite ).

Hint: Consider c: = inf&#945; c&#945; and d: = sup&#945; d&#945; and show that I = ( c, d ).

Solution contains detailed step-by-step explanation.

\$2.19