We reformulate a key definition given by Wáng and Ågotnes [18] to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of (varphi ) to be the "local truth" of (varphi ), semantically rendered via an interior operator. This is closely related to the notion of (varphi ) being "knowable". We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness.