This is somehow connected to this question.

I can think of at least four topologies to put on $C_c(M)$:

1. Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\infty$ topology on $C^{\infty}(M)$.
2. Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the strong Whitney $C^\infty$ topology on $C^{\infty}(M)$.
3. Topologize $C^{\infty}_c(M)=colim_{K\subseteq M} C^{\infty}_K(M)$ as direct limit over compact subsets $K\subseteq M$ where $C^{\infty}_K(M)$ are the functions, which have support in $K$ with the Whitney topology. (The strong and the weak one should coincide in that case.)
4. Take $h\in C^{\infty}_c(M)$ a smooth function $\epsilon\colon M\rightarrow (0,\infty)$, vector fields $X_1,...,X_k$ and declare the subsets of the shape $\{g\in C^{\infty}_c(M)\colon\text{ }|X_1..X_k(h-g)(x)|<\epsilon(x)\forall x\in M\}$ as a subbase varying over $h,k\in\mathbb{N}_0, X_1,...,X_k$ and $\epsilon$.

How are these topologies related?

My vague guesses are:

• Topology 2.) is finer than topology 1.), but those are in general not equal.
• Topology 2.) is finer than topology 3.), but in general not the same.

I have no idea how to relate topology 4.) to the others.

This is somehow connected to this question.

I can think of at least four topologies to put on $C_c(M)$:

1. Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\infty$ topology on $C^{\infty}(M)$.
2. Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the strong Whitney $C^\infty$ topology on $C^{\infty}(M)$.
3. Topologize $C^{\infty}_c(M)=colim_{K\subseteq M} C^{\infty}_K(M)$ as direct limit over compact subsets $K\subseteq M$ where $C^{\infty}_K(M)$ are the functions, which have support in $K$ with the Whitney topology. (The strong and the weak one should coincide in that case.)
4. Take $h\in C^{\infty}_c(M)$ a smooth function $\epsilon\colon M\rightarrow (0,\infty)$, vector fields $X_1,...,X_k$ and declare the subsets of the shape $\{g\in C^{\infty}_c(M)\colon\text{ }|X_1..X_k(h-g)(x)|<\epsilon(x)\forall x\in M\}$ as a subbase varying over $h,k\in\mathbb{N}_0, X_1,...,X_k$ and $\epsilon$.

How are these topologies related?

My vague guesses are:

• Topology 2.) is finer than topology 1.), but those are in general not equal.
• Topology 2.) is finer than topology 3.), but in general not the same.

I have no idea how to relate topology 4.) to the others.