Definition:Open Refinement

Definition

Let $T = \struct {X, \tau}$ be a topological space.

Let $\UU$ and $\VV$ be covers of $X$.

Then $\VV$ is an open refinement of $\UU$ if and only if:

$(1): \quad \forall V \in \VV: \exists U \in \UU: V \subseteq U$

$(2): \quad \VV \subseteq \tau$

That is:

$(1): \quad \VV$ is a refinement of $\UU$

$(2): \quad$ All elements of $\VV$ are open in $T$