I'm using Cc(X) to mean the 'space of continuous complex-valued functions on X which have compact support'. Naturally I'm assuming X is locally compact. I'm using the usual uniform norm and point wise multiplication.
I've got "X is compact implies Cc(X) is a Banach algebra". I don't think the converse, ie "Cc(X) is a Banach algebra implies X is compact" holds, but I'm struggling to understand completely why.
I've seen the counterexample of X being the first uncountable ordinal which is locally compact but Cc(X) is complete, although I'm not sure if this is a strong enough counterexample in my case.
Any help would be appreciated, thanks!