# \$X\$ is a \$T_2\$ space. Let \$fin C_0(X), gin C_c(X)\$. Prove that, \$fgin C_c(X)\$.

Here $$C_0(X)$$ denotes the collection of all such continuous maps $$f:XtoBbb{C}$$ such that $$forall epsilon>0 exists$$ compact set $$K$$ such that $$|f(x)|.

And $$C_c(X)$$ denotes the collection of all such continuous maps $$g:XtoBbb{C}$$ such that $$text{supp}(g):=overline{{xin X|g(x)ne 0}}$$ is compact.

It’s easy to see that $$C_c(X)subset C_0(X)$$.

Suppose $$fin C_c(X)$$. Then $$text{supp}(fg)subset text{supp}(f)cuptext{supp}(g)$$ and since union of two compact sets is compact and $$text{supp}(fg)$$ is closed, $$text{supp}(fg)$$ is compact, hence $$fgin C_c(X)$$.

But I’m stuck with the case $$fin C_0(X)setminus C_c(X)$$?

Can anyone help me in this regard? Thanks for your help in advance.