Today I give you a mathematical pleasure, which you might want to take up and measure (What am I?)

Today I give you a mathematical pleasure,

which you might want to take up and measure.

With a pole, or a hole, oh well,

you might want to read up on Borel.

My spirit can always be found in myself,

not somewhere else way up on a shelf.

With cups and caps you can bet,

that I'm nothing more than a well-equipped set.

Hint 1

No calculations, just math .

Hint 2

A hole or a pole is an allusion to topological spaces.

Hint 3

Let's just say that BSA, is not bovine serum albumin.

You are, specifically,

the Borel $$\sigma$$-algebra on a topological space.

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.

• V jnf guvaxvat bs n Obery fcnpr nf gur nafjre (onfvpnyyl gur fnzr guvat), ohg V pna'g cynpr gur ubyrf naq cbyrf guvat. Vg'f znxvat zr guvax bs mrebrf naq cbyrf bs pbzcyrk shapgvbaf, ohg V pna'g chg gung pyhr vagb vg. – Anon Apr 25 at 6:18
• You're so close! The thing you mention as the answer is not necessarily closed under intersection. I've added a hint. – Galen Apr 25 at 14:42
• @Galen OK, edited to a new answer. (I actually thought of this first - I know topology much better than measure theory - but then the mention of Borel sent me in the other direction.) – Rand al'Thor Apr 25 at 16:06
• @Galen A topological measure space then? – Rand al'Thor Apr 25 at 16:10
• @Galen Thank you for making a mathematical riddle! Btw, you might like this one (shameless plug). – Rand al'Thor Apr 25 at 16:19