Suppose you have a matroid, and $T$ is a subset of a spanning set $S$.


Now consider the contraction of the matroid to the set $T$ and suppose $X$ is a spanning subset of $T$ with respect to that matroid structure.


Is $(S\setminus T)\cup X$ a spanning set of the original matroid?



Extending spanning sets on contractions of matroids
No. If the complement $\bar{S}$ of $S$ is also spanning, then the contraction to $T$ is trivial and $X=\emptyset$ is spanning. But $S\setminus T$ is not necessarily spanning (it may be even empty: take $T=S$).


