Brutkey

John Carlos Baez
@johncarlosbaez@mathstodon.xyz

@MartinEscardo@mathstodon.xyz - in real analysis, the definition of 'Lebesgue integration' is incredibly important. This relies on the definition of Lebesgue integrable function f:ℝ→ℝ, and thence on the definition of measurable function, and thence on the definition of Lebesgue measurable set, and thence on the definition of Borel set. But for most analysts, integration is the main point of all those more fundamental definitions.

In functional analysis and quantum physics, I'd say the definition of 'Hilbert space' is incredibly important!

In algebraic geometry, I'd say the definitions of 'ample line bundle' and 'coherent sheaf' are incredibly important.

In homotopy theory, the definition of 'simplicial object' is incredibly important.

I'm picking definitions where it's not obvious right away why those definitions will turn out to be so central.