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A collection $C$ is called an open cover if it is a cover of $X$ and every member of $C$ is an open set in the topology $\tau$. Showing $\bigcup C = X$ only proves it is a cover; membership in $\tau$ proves it is an open cover.

Conditions: Every member of $C$ belongs to $\tau$.; $\bigcup C = X$.

A collection $C$ is called an open cover if it is a cover of $X$ and every member of $C$ is an open set in the topology $\tau$. Showing $\bigcup C = X$ only proves it is a cover; membership in $\tau$ proves it is an open cover.

Conditions: Every member of $C$ belongs to $\tau$.; $\bigcup C = X$.