Re: Can we say (0, 1) is compact in (0, 1)?


comtech wrote:
> Hi all,
>
> Suppose my metric space M is (0, 1),
>
> and set A is (0, 1) in M,
>
> can we say that A is compact (in M)?
>
> My reasoning is that:
>
> A is both closed and open in M, since A=M, let's deem it as closed;
> A is bounded too,
> So A is bounded and closed => A is compact, by the Heine-Borel
> theorem...
>
> Am I correct? I am really not sure...

The Heine-Borel Theorem holds for subspaces of R^n for all n, but says
nothing about other metric spaces, such as (0, 1). As it turns out, it
can be shown that (0, 1) is not compact in (0, 1) the same way that it
can be in R. (0, 1) is the union of (1/n, 1 - 1/n) where n ranges
from 3 to infinity. Each (1/n, 1 - 1/n) is open in (0, 1) since it is
open in R, and clearly no finite collection of these intervals can
cover (0,1).

Mike

.

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