# Baire theorem explanation

Can someone give an intuitive or geometrical explanation of the Baire category theorem?

Thank you in advance!

Theorem (Baire): Let $(X,d)$ be a complete metric space. For every sequence $\{O_n\}_{n\in\mathbb{N}}$ of open and dense subsets of $X$ the $\bigcap$$_{n\in\mathbb{N}}$$O_n$ is a dense subset of $X$

A set is in a metric space $(X,d)$ is of $second$ $category$ if it is not a union of nowhere dense subset of $X$

Corollary: Every complete metric space is of second category

• The usual proof gives much intuition if you were to draw the usual balls on paper and see they will get smaller and smaller but all contain a common point. – Hasan Saad Sep 11 '16 at 12:10
• Thank you. But is there a more general geometric explanation of this theorem? – Marios Gretsas Sep 11 '16 at 12:12
• I dont think I have one. All that I can see is that because of this completeness the point they "shrink at" is actually in this space and they dont shrink at nothing. For example the rationals can shrink to nothing (a neighborhood of rational points around some irrational number). Thats all I can see from Baires category theorem but the others probably have more. – Hasan Saad Sep 11 '16 at 12:17
• I don't think there is an easy way of understanding the underlying concept. My guess is that Baire was as surprised himself when discovering this, I tried to find his original 1899 paper, but couldn't. Anyway, once you accept and digress, it becomes a very useful tool. – H. H. Rugh Sep 11 '16 at 12:19
• I recommend Brian Scott's explanation on this question: math.stackexchange.com/questions/2010948/… – rem Mar 20 at 23:24