**Donal O'Regan**

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland; donal.oregan@nuigalway.ie

Received: 8 January 2020; Accepted: 18 February 2020; Published: 25 February 2020

**Abstract:** A simple theorem is presented that automatically generates the topological transversality theorem and Leray–Schauder alternatives for weakly upper semicontinuous, weakly compact maps. An application is given to illustrate our results.

**Keywords:** weakly upper semicontinuous; essential maps; homotopy
