in complex geometry there is Oka's principle. It says that on a stein manifold, if there is no obstruction to a continuous construction there will be no obstruction to a holomorphic construction either.
[A stein manifold is a holomorphic submanifold of C^n : it is the analogue in the complex analytic category of affine varieties.]
The simplest example: consider an open subset U of C and a point p in U. Given a holomorphic function on U-p, it can be holomorphically extended across p as soon as it can be continuously extended ( U is stein, as is every connected non-compact complex manifold of dimension one: Behnke-Stein's theorem).
Next simplest example: on a stein manifold two holomorphic line bundles are analytically isomorphic as soon as they are topologically isomorphic.
This last result was vastly generalized by Grauert to the case of an arbitrary principal bundle( with group an arbitrary complex lie group) over a stein manifold: a remarkable achievement.
And Grauert's theorem has been generalized by Gromov to a new instance of Oka's principle...
You will find a very poetic homage to Oka and his principle in the following article
in which J.Kajiwara explains that Oka lives in Buddha's paradise.
PS Here is a very exhaustive survey by Pit-Mann Wong I have just found by googling