Definition:Open Refinement

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Let $T = \left({X, \tau}\right)$ be a topological space.

Let $\mathcal U$ and $\mathcal V$ be covers of $X$.

Then $\mathcal V$ is an open refinement of $\mathcal U$ iff:

$(1): \quad \forall V \in \mathcal V: \exists U \in \mathcal U: V \subseteq U$
$(2): \quad \mathcal V \subseteq \tau$

That is:

$(1): \quad \mathcal V$ is a refinement of $\mathcal U$
$(2): \quad$ All elements of $\mathcal V$ are open in $T$