if it has a countable base for its topology. Second-countable spaces are always separable, first-countable and Lindelöf. Connectedness Connected. A space is connected if it is not the union of a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself. Locally connected. A space is locally connected if every point has a local base consisting of connected sets. Totally disconnected. A space is totally disconnected if it has no connected subset with more than one point. Path-connected. A space X is path-connected if for every two points x, y in X, there is a path p from x to y, i.e., a continuous map p: [0,1] → X with p(0) = x and p(1) = y. Path-connected spaces are always connected. Locally path-connected. A space is locally path-connected if every point has a local base consisting of path-connected sets. A locally path-connected space is connected if and only if it is path-connected. Arc-connected. A space X is arc-connected if for every two points x, y in X, there is an arc f from x to y, i.e., an injective continuous map f: [0,1] → X with p(0) = x and p(1) = y. Arc-connected spaces are path-connected. Simply connected. A space X is simply connected if it is path-connected and every continuous map f: S1 → X is homotopic to a constant map. Locally simply connected. A space X is locally simply connected if every point x in X has a local base of neighborhoods U that is simply connected. Semi-locally simply connected. A space X is semi-locally simply connected if every point has a local base of neighborhoods U such that every loop in U is contractible in X. Semi-local simple connectivity, a strictly weaker condition than local simple connectivity, is a necessary condition for the existence of a universal cover. Contractible. A space X is contractible if the identity map on X is homotopic to a constant map. Contractible spaces are always simply connected. Hyperconnected. A space is hyperconnected if no two non-empty open sets are disjoint. Every hyperconnected space is connected. Ultraconnected. A space is ultraconnected if no two non-empty closed sets are disjoint. Every ultraconnected space is path-connected. Indiscrete or trivial. A space is indiscrete if the only open sets are the empty set and itself. Such a space is said to have