> > > > Is this the same as neighbourhood space defined as follows. > > > > > > > > (S, N), S a set, N a map S -> PPS (P for power set) and > > > > i) x in S => N(x) =/= 0 > > > > ii) x in S, M in N(x) => x in M > > > > iii) x in S, M in N(x) => (L superset M => L in N(x) > > > > iv) x in S, L, M in N(x) => L intersect M in N(x) > > > > v) x in S, M in N(x) => exists L in N(x) s.t. > > > > L subset M and, forall y in L, L in N(y) > > > > ? > > > I wrote of my axioms just for reasons of euphony, I think they are > > > Hausdorff's. Again, someone more knowledgeable than I will know. > > > > Here's from the web. Are these equivalent to yours? > > > > The axioms formulated by Hausdorff (1919) for his concept of a > > topological space. These axioms describe the properties satisfied > > by subsets of elements x in a neighborhood set E of x . > > > > 1. There corresponds to each point x at least one neighborhood > > U(x) , and each neighborhood U(x) contains the point x . > > > > 2. If U(x) and V(x) are two neighborhoods of the same point x , > > there must exist a neighborhood W(x) that is a subset of both. > > > > 3. If the point y lies in U(x) , there must exist a neighborhood > > U(y) that is a subset of U(x) . > > > > 4. For two different points x and y , there are two corresponding > > neighborhoods U(x) and U(y) with no points in common. > > Mine doesn't imply the separation axiom 4.
Without 4, are they equivalent? If so, then how would you prove from 1-3, your axiom iii) x in S, M in N(x) => (L superset M => L in N(x) ?