The Connectivity and Local Connectivity of Several Special Pre-topological Spaces

The connectivity and local connectivity of several special pre-topological spaces, such as the discrete pre-topological space with more than two points, the parity pre-topological space and the interval sets pre-topological space, were put under examination. Then the notion of the ordered pre-topological space was established, and the lexicographical ordered pre-topological space was proved to be connected and, to be exact, locally connected.