Should the cold water supply be constantly open for hot water heating system?
Closed.

Two reasons - main one, you need to know if it's leaking. A properly functioning system should not need "make up water" on a regular basis.

The other, related reason - if the system is leaking and constantly taking in new water, it's also taking in new dissolved oxygen in that water - and 99.9% of hot water heating systems are built with a mixture of copper and iron parts (or all iron parts) that will rust terribly if continuously supplied with oxygen. A non-leaking system rusts out what oxygen is in the water, and then stops rusting. Every time you add new water it rusts more; so the less you add new water, the less the interior of the system rusts.