Open tubes
In Chapter 31, “The Compact-Open Topology”, it is asserted that “clearly” the third topology on (with
and
the real line) is finer than the second. The issue is more fully discussed here.
In this post I discuss the concept of open tubes, particular subsets of the Cartesian product with
a set and
a metric space.
Definition
Let be a set and
a metric space, with distance
.
For any and any
, we note
the open ball of
of radius
and centred on
, that is set of all
such that
. Open balls are notoriously open subsets.
Let be any mapping
, and
a real
.
Then the open tube of radius
and centred on
is the following subset of
:
Equivalently:
The “open” in “open tube” refers to its being based on open balls (second expression above), but does not imply that it is itself necessarily open; indeed, we have not given any topology to , so we don’t have any particular one on
and cannot speak of
being open or not.
However, if is a topological space and
is continuous
, then
is open, as I will show.
An open tube centred on a continuous mapping is open
If
is a topological space and
is a continuous mapping
, then any open tube centred on
is an open subset of
.
Proof:
I will show that in the above circumstances is a neighbourhood of each of its points.
Let be an eventual point of
.
Then .
Lettuce define . We have
.
Let . Since
is continuous and
is an open subset of
,
is an open subset of
. Since
,
.
Let , open subset of
. Clearly,
, so
is a neighbourhood of
.
Let be an eventual point of
.
, hence
, that is
.
, hence
.
Since and
, this leads to:
With:
Thus we have: .
Which implies: .
Hence .
Thus we have found a neighbourhood of
that is a subset of
, which implies that
itself is a neighbourhood of
.
Thus is an open subset of
.