X-Nico

Compact space

Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt.

Generalized flag variety

In the smooth setting, generalized flag manifolds are the compact flat model spaces for Cartan geometries of parabolic type, and are homogeneous Riemannian manifolds under any maximal compact subgroup of G.

Global section functor

Let $\backslash mathcal\; O\; X$ be the sheaf of holomorphic functions on the compact connected complex manifold X, then by the maximum principle, global sections are constant, ie. $\backslash Gamma\; (X,\; \backslash mathcal\; O\; X)\; =\; \backslash mathbb\; C$

Nagata's compactification theorem

Deligne showed, in unpublished notes expounded by Conrad, that the condition that S is Noetherian can be replaced by the condition that S is quasi-compact and quasi-separated.

Zorn's lemma

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

see also

Grothendieck space

Grothendieck spaces which are not reflexive include the space C(K) of all continuous functions on a Stonean compact space K, and the space L^∞(μ) for a positive measure μ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).