Locally operator

Definition

Let $p$ be a property of topological spaces. The property locally $p$, is defined as the property of being a topological space in which every point has an open neighbourhood, which, in the subspace topology, satisfies $p$.

Facts

• For any property $p$ of topological spaces, $p$ implies locally $p$.
• There is a variant of the locally operator, called the strongly locally operator, and the prefix adjective locally is used for either of these. The strongly locally operator requires that for any neighbourhood of a point, there exists a smaller neighbourhood whose closure lies within the given neighbourhood, which has the required property. If $p$ is hereditary on open subsets, a regular is locally $p$ iff it is strongly locally $p$.