Let \(\Gamma\) be a cover for \(X\). It is called fundamental cover of \(A\) if for each \(A\subset X\): \( A\cap B\) is open in \(B\) for all \(B \in \Gamma\) implies \(A\) is open in \(X\). Somehow the open cover gives the converse of our remark concerning relative topology. It suffices to check the openness at the level of each subset in the cover elements to state the openness at the level of the whole space. Similarly, each subset \(A\) of \(X\) such that \(A\cap B\) is closed in \(B\) for all \(B \in \Gamma\) is closed itself. Covers that admit fundamental refinement is also fundamenral.


All open covers, and all finite or locally finite closed covers are fundamental. In metric space, the collection of all open balls is a fundamental cover. Therefore, in order to check if a set \( A \) is open it suffices to check that \(A\cap B_r(x)\) is open for all \(r>0\) and for all \( x\in X \). Since \(A \) is union of open balls,