Some characterizations of COTS with endpoints

We prove that a connected space X is a COTS with endpoints iff there is a one–one Darboux function from X onto a space with endpoints. Using this result, we show that a connected space X   is homeomorphic to the closed unit interval if it is T1T1 separable and locally connected and there is a one–one Darboux function from X onto a space with endpoints. Also we obtain some other characterizations of COTS with endpoints and some characterizations of the closed unit interval.