The following topological property arose in the context of my friend Don Hadwin's operator theory research, and he asked me to ask here if the property occurs in the literature and has a name.

Property of a space $X$: given $n$ open sets in $X$, there exists a single path that meets them all.

I offered the following filter-theoretic paraphrase. Observe that the neighborhood basis of each point in $X$ determines a filter base on the set of path component of $X$ (where each neighborhood corresponds to the set of path components it meets).

Property of a space $X$: The union of all these filter bases generates a non-trivial filter.

Since Don asks for a path, not necessarily an arc, that meets the open sets, the order of the open sets will not matter. That said, the arc version of his property indeed comes in two flavors, depending upon whether order matters or not (the real line will enjoy one property but not the other). It would be interesting to know if either arc version has appeared in the literature.

Finally let me point out that a space with a dense path component will enjoy Don's property, but one can cook up a subset of R^2 to show that the converse fails. So "has a dense path component" doesn't do.

Please feel free to mention any closely related property in the literature that seems relevant.