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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 $\tau$.; $\bigcup C = X$.

To check whether a collection $C$ is a cover of $X$, form the union of all members of $C$. If the union equals $X$ (or contains $X$), then every point of $X$ lies in at least one member of $C$, so $C$ is a cover.

Conditions: $C$ is a collection of subsets of $X$.; The test is whether $\bigcup C = X$.

To check whether a collection $C$ is a cover of $X$, form the union of all members of $C$. If the union equals $X$ (or contains $X$), then every point of $X$ lies in at least one member of $C$, so $C$ is a cover.

Conditions: $C$ is a collection of subsets of $X$.; The test is whether $\bigcup C = X$.

To find a finite subcover, select a finite subcollection $C_1 \subseteq C$ such that the union of the sets in $C_1$ still equals $X$. The chosen sets must come from the original cover $C$.

Conditions: $C_1$ must be a subcollection of $C$.; $C_1$ must be finite.; The union of $C_1$ must equal $X$.

To find a finite subcover, select a finite subcollection $C_1 \subseteq C$ such that the union of the sets in $C_1$ still equals $X$. The chosen sets must come from the original cover $C$.

Conditions: $C_1$ must be a subcollection of $C$.; $C_1$ must be finite.; The union of $C_1$ must equal $X$.