Interactive Real Analysis
- part of Examples 5.2.5(b):
Let S = [0, 1]. Define
=
{ t 
R :
| t - 
| <
and
S}
for a fixed > 0.
Is the collection of all
{ },
S,
an open cover for S ? How many sets of type
are actually needed to cover S ?
Back
First, each set
=
{ t R :
| t - | <
and
S}
for a fixed > 0.
Is the collection of all
{ },
S,
an open cover for S ? How many sets of type
are actually needed to cover S ?
Back
is an open set, because it is the same as an interval around
of length
2 .
Second, the union of all sets
equals the open interval
(- ,
1 + ),
so it contains the set S. Therefore, the collection
{ },
S
is an open cover of S.
The collection
{ },
S
consists of uncountable many sets. In order to cover S,
however, we need only a finite subcollection for any given
.
To see this, fix an
> 0.
Then let N be the smallest integer greater than
1 / ,
and define
=
k * , k = 0, 1, 2, ... N
},
k = 0, 1, 2, ..., N
is a covering of S. That is, this new collection forms a
finite subcover of S with respect to the original collection
of sets.