Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
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In mathematical analysis, the Whitney covering lemma asserts the existence of a certain type of partition of an open set in a Euclidean space.
Grothendieck spaces which are not reflexive include the space C(K) of all continuous functions on a Stonean compact space K, and the space L∞(μ) for a positive measure μ (a Stonean compact space is a Hausdorff compact space in which the closure of every open set is open).