Connected topological space pdf

J, and let A be this countable subset of X. Now, we intend to prove that A satisfies the assertion of the fundamental lemma. Note that this necessary condition of strong separability is a slight generalization of the one encountered in classical utility theory [see, e. The following proposition points out some relationship between strong separability and the different continuity conditions: Proposition. Proof Let us show that c, x is open for each x in X. Let us assume that c,x is not empty, otherwise the result is immediate.

Necessary and sufficient conditions for the existence of a continuous representation, when X, z is connected Theorem. M oreover, when such a representation exists, there exists one such that u and v are utility functions for, respectively , the complete preorders 5 1 and sz.

Proof The necessary part has already been proved. Let us now turn to the sufficiency part. Let us first show that the complete preorder s1 is perfectly separable. It remains to be shown that v is continuous on X,z. Assume that a E [O, l , otherwise the result is immediate.

Hence E, is open. Let x0 E F,. In the particular case where - is transitive it may be worth noticing that the conditions of the above theorem are nothing else but the usual ones which are necessary and sufficient for the existence of a continuous utility function on a connected and completely preordered space. This clearly points out the intimate connection between our results and those of classical utility theory. Proof: The necessary part has already been proved see Section 3.

They are density conditions very similar to those of our fundamental lemma, and to those achieved independently by Doignon, Ducamp and Falmagne Then, by merely adding necessary conditions of continuity such as strong pseudo-transitivity [see Bridges a ], it might be possible to solve the problem of the existence of a continuous representation in the general case.

If the results prove to be of interest, they will be related in a future paper. Chateauneuf, A preference relation on a connected topological space References Birkhoff, G.

Bridges, D. Cohen, M. Debreu, G. Doignon, J. Ducamp and J. Falmagne, , On realizable biorders and the biorder dimension of a relation, Journal of Mathematical Psychology, Vol. Fishburn, P. Jaffray, J. Lute, R. Peleg, B. Scott, D. Suppes, , Foundational aspects of theories of measurement, Journal of Symbolic Logic 23, Related Papers Representing preferences with nontransitive indifference by a single real-valued function By Gianni Bosi.

From local to global additive representation By Peter Wakker. Numerical representations of interval orders By Gianni Bosi. Preliminaries Definition 2. Definition 2. Primary 54A05, 54A10; Secondary Keywords. Modak, T. Theorem 2. Hence from Theorem 2. This is obvious from the above diagram. By Theorem 2. For converse of Theorem 2. Example 2. This is a contradiction. Theorem 3. The proof is similar with Theorem 3. By Theorem 3. Thus G is an empty set. Since G is nonempty, this is a contradiction.

Corollary 3. This implies that G is empty. By the similar way, we have that H is empty. Thus by Theorem 3. By the similar way, we obtain that B is open. Definition 3. Suppose A and B intersect. Then, by Theorem 3. Thus A and B are disjoint. Let A be a proper subset of C. This completes the proof. References [1] J. Dontchev, M.

Ganster, D. Rose, Ideal resolvability, Topology Appl. Ekici, T. Noiri, Connectedness in ideal topological sapces, Novi Sad J. Hayashi, Topologies defined by local properties, Math.