On the continuous analogue of the Szpilrajn Theorem I

One of the best known theorems in order theory, mathematical logic, computer sciences and mathematical social sciences is the Szpilrajn Theorem which states that every partial order can be refined to a linear order. Since in mathematical social sciences one frequently is interested in continuous linear orders or preorders, in this paper the continuous analogue of the Szpilrajn Theorem will be discussed.