As the communications interface was
As the communications interface was developed by two separate organisations with differing software processes and tools, a modular approach to the safety case was deemed appropriate. The modular approach related the communications interface safety argument to the safety arguments for both the ACS and IMPS, as they all contribute to the overall Vessel Safety Case. The modular approach to creating the safety argument began with creating modules in the safety case for each of the software modules in the software architecture, then defining the top-level argument that is to be justified. The modular top-level argument for the case study is shown in Fig. 12. The ‘IFSafe’ module is the contract module (based on the GSN extensions discussed by ), and a Safety Case Contract was used to record the dependencies that existed between the safety argument modules (within the figure, for brevity, only the contract between the IMPSSafe and IFSafe modules is shown). The organisation responsible for overseeing these contracts was the shipbuilder.
Further detail of the safety case is given in .
Discussion and evaluation
Introduction Topological spaces are assumed to be connected for any consideration of cut points. The concept of COTS (=connected ordered topological space), defined by Khalimsky, Kopperman and Meyer , does not require any separation axiom. In view of the applications of cut points (see e.g. ) and the fact that many connected topological spaces used for cut points like the Khalimsky line, are not , the assumption of separation axioms is avoided as far as possible. Since by Theorem 3.4 of , connected topological spaces have at least two non-cut points, it follows from Remark 4.5 of  that such topological spaces with at most two non-cut points turn out to be COTS with endpoints. It is shown in  that a connected topological space with endpoints is a COTS with endpoints. It is proved in  that a connected topological space is a COTS with endpoints iff it admits a continuous bijection onto a topological space with endpoints. Papers  and  introduce several euk 134 sale of spaces, whose members are COTS with endpoints. In this paper, we find some characterizations of COTS with endpoints. Notation and definitions are given in Section 2. The main results of the paper appear in Section 3. In Section 3, we prove that a connected space X is a COTS with endpoints iff there is a one–one Darboux function (a function which takes connected sets onto connected sets, , ) from X onto a space with endpoints. Using this result, we prove that a connected space X which is separable and locally connected is homeomorphic with the closed unit interval if 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 with the concept of .
Notation and definitions Notation and definitions of the current paper mainly follow the papers  and . Most of them are included in this section for the sake of completeness. Let X be a space. For denote the closure of H in Y. X is called  if every open cover of X has a finite subcollection such that the closures of the members of that subcollection cover X. A filter base γ on X is said to be fixed if . An o-filter base on X is a filter base whose members are open subsets of X. An o-filter base γ is a regular o-filter base if each member of γ contains the closure of some member of γ. X is called  if every regular o-filter base on X is fixed. A subset Y of X is if Y is as a subspace of X. X is called  if every singleton set is either open or closed. A point called a cut point if there exists a separation of . is used to denote the set of all cut points of X. Let . A separation of is denoted by if the dependence of the separation on x is to be specified. is used for the set . For and a separation of , if Y is a connected subset of , then is used to denote the separating subset of containing Y. is used for the set . If , we write for , for . Let . A point , is said to be a separating point between a and b or x separates a and b if there exists a separation of , with and . is used to denote the set of all separating points between a and b. For , we shall write for a separation of . If we adjoin the points a and b to , then the new set is denoted by . Call a cut point convex set if for , . A space X is called a space with endpoints if there exist a and b in X such that .