Can you keep your web space?

Yes, if the web space is independent of your current provider.

Yes, for free during 6 months after your contract with your current provider has ended, if you ask this before the end of your current contract and if that web space was made available by your current provider. The Internet providers have concluded an agreement in that sense. 

Your current provider should inform you exactly and indicate once a year on your bill how you can keep your web space.