Assume DC($\aleph_1$).

 Can we define the following: 

 1. Basis for a vector space $V$ over a field $K$ such that $\operatorname{card}(K) \leq \aleph_1$ and we happen to find a generating set of $V$ of cardinality $\leq \aleph_1$. 
 2. Linear dimension making the same assumption as above.
 3. Transcendence degree of a field $K$ over $\mathbb{Q}$ if we happen to know that $K$ is of cardinality $\leq \aleph_1$.
 etc...