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The example demonstrates that the chosen open cover $C$ of the finite space $X$ has a finite subcover $C_1$. This illustrates the core property used in the definition of compactness: every open cover admits a finite subcover.

Conditions: $X$ is the space under consideration.; $C$ is an open cover of $X$.; $C_1$ is a finite subcollection of $C$ covering $X$.

The example demonstrates that the chosen open cover $C$ of the finite space $X$ has a finite subcover $C_1$. This illustrates the core property used in the definition of compactness: every open cover admits a finite subcover.

Conditions: $X$ is the space under consideration.; $C$ is an open cover of $X$.; $C_1$ is a finite subcollection of $C$ covering $X$.

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 $X$.