You are, specifically,
the Borel $\sigma$-algebra on a topological space.
(Iterations of this answer:
I got immediately the general theme, but I put $\sigma$-algebra at first, then changed to topological space after the second hint was added, then finally put them together in the intended way after the third hint.)
Today I give you a mathematical pleasure,
which you might want to take up and measure.
Every topological space can be turned into a measurable space by using the Borel $\sigma$-algebra.
With a pole, or a hole, oh well,
Topology is about deformation of objects, but a hole cannot be deformed away.
you might want to read up on Borel.
Borel sets again.
My spirit can always be found in myself,
not somewhere else way up on a shelf.
A topology on a set contains the set itself as one of its elements, and so does a $\sigma$-algebra.
With cups and caps you can bet,
that I'm nothing more than a well-equipped set.
A topology is closed under certain kinds of union and intersection (cup and cap), and it's simply something that a set can be equipped with. Same for a $\sigma$-algebra.