## 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 .