Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general?

If no, is there a well-known condition of when they exist?

If yes, how can I describe the topology of the colimit?

(The example in my mind has following properties. First, it is a filtered colimit; it is a chain of inclusions, not necessarily linearly ordered, and not necessarily countable. Also, each inclusion is in general not a homeomorphism onto the image. Topology gets coarser and coarser as you embed into a larger space.)

(I'm also interested on a condition when a subset of the colimit is contained in one of the component spaces. For example, in the test function space $\mathcal{C}_c^\infty(\mathbb{R}^n)$ (which is of course NOT a colimit in the category of topological vector spaces), any bounded set is contained in one of $\mathcal{C}^{\infty}(K)$, $K$ a compact set. When can I say something similar?)