While topology given by a linear order has been extensively studied, this cannot be said about the case when the order is given only locally. The aim of this paper is to fill this gap. We consider relation between local orderability and separation axioms and give characterisation of those regularly locally ordered spaces which are connected, locally connected or Lindel¨of. We prove that local orderability is hereditary on open, connected or compact subsets. A collection of interesting examples is also offered.
