||I would like to introduce the join operator. If $X$ and $Y$
are smallest-neighborhood spaces, i.e., topological spaces where
each point has a smallest neighborhood, we can form a new topological
space, the join of $X$ and $Y$. Conversely, a space can sometimes be
decomposed as a join of simpler spaces.
I shall try explain the intuition behind the join operator, and
why I have found it useful in my work on digital manifolds.
To the best of my knowledge, this operator is not described in the
It is my belief that the ideas presented can be used to give simple proofs
of the (well-known) facts that (for example) the 8-connected plane and the
hexagonal grid are not topological spaces. Hopefully, I will be able to
provide such proofs on Monday.