# How to prove that the affine hull of a set is a closed set.

I am asked to prove that $$\text{aff}(X)$$ is closed in the topological sense. I found some posts where people show that it is closed under affine combinations (what it's quite obvious because the definition of affine set).

So i want to know if this prove is equivalent to show that $$\text{aff}(X)$$ is closed in the topological sense. In other case, I would like to know how to prove it.

• In general , the claim is false. The vector space of polynomials is an affine subset of the vector space of formal power series, but not closed. – Hagen von Eitzen Feb 14 at 7:00
• @HagenvonEitzen In en.wikipedia.org/wiki/Affine_hull it is also a propertie, what surprise me harder – Lecter Feb 14 at 7:17
• @Lecter: Wikipedia discusses the case $X \subset \mathbb R^n$. In this case, any affine subspace is closed. In the infinite dimensional case, this is no longer true. It is also worthwhile to rephrase your question as: "Is every affine subspace topologically closed?", since every affine subspace is its own affine hull (obviously). – gerw Feb 14 at 7:47