I'm trying to figure Whitnney topology out. For example I need to prove that $C^0(\mathbb{R},\mathbb{R})$ is path connected in Weak Whitney topology. I think that path from $f$ to $g$ is $l(t)=gt+f(1-t)=t(g-f)+f$, $t∈[0,1] $. But how to prove that it is a path in $C^0(\mathbb{R},\mathbb{R})$ in WWT?
The definition of WWT from M.Hirsch "Differential topology"
A weak topology on $C^r(M,N)$ is generated by the sets defined as follows. Let $f∈C^r(M,N)$. Let $(φ,U),(ψ,V)$ be charts on $M,N$; Let $K⊂U$ be a compact set such that $f(K)⊂V$; let $0<ϵ≤∞$.
Define a weak subbasic neighborhood $$N(f;(φ,U),(ψ,V),K,ϵ)$$ to be the set $C^r$ maps $g:M→N$ such that $g(K)⊂V$ and
$$∥D^k(ψfφ^{−1})(x)−D^k(ψgφ^{−1})(x)∥<ϵ$$ for all $x∈φ(K),k=0,…,r$. This means the local representation of $f$ and $g$, together with their first $k$ derivatives, are within $ϵ$ at each point of $K$.
The weak topology on $C^r(M,N)$ is generated by these sets; It defines a topological space $C^r_W(M,N)$. A neighborhood of $f$ is thus any set containing the intersection of a finite number of sets of this type.