No, because excluding the first condition from the definition of topology, what we obtain can not more be called a topology !!! All three conditions are equally important!

For example, if we did not have condition 1 in the definition, then we would not be able to work in topological spaces with notion called connectedness.

of course, no, because this is a first condition in definition of topology. In this case you make another structure and depending to you for named.

Dear Shanker,

Note that axiom 1 is an immediate consequence

of axioms 2 and 3. Namely, the emptyset is a

finite subset of the family tau.

Otherwise, from the theory of relator ( generalized

uniform) spaces, you can see that minimal structures,

generalized topolgies and stacks are the most important

families of sets in " Topology".

In Algebra and Analysis , it is necessary to consider

several further important families of sets. For instace,

ideals, semirings and sigma-algebras.

Otherwise, after the discoveries of Pervin (1962),

Hunsaker-Lindgren (1970), Fletcher-Lindgren (1982) and

several others, including me (1987, 1992, 2002

and 2007) too, only stupid people are dealing with

generalized topologies, closures, proximities and

convergences without using generalized uniformities.

With best regards,

Arpad Szaz

Yes I do agree with that.

Dear Teodorescu and Shanker,

Thank you very much for recommending and accepting

my very coarse and impolite, but true statements.

With best vishes,

Arpad Szaz

Dinu Teodorescu,

I agree with that answer

Hi everybody,

Assume that such a "topology" is defined on a set $X$.

1. What would be the closed sets ?

- The complements in $X$ of open sets ? Why ? $X$ is not open.

- The complements of open sets in $O$, the union of all open sets. But then this reduces to a topology on $O$.

2. How to define separation axioms for elements not in $O$ ?

3. If $X$ is the target set of a function with image in $X \setminus O$, how to define continuity. Even constant functions would fail to be continuous.

etc.

No, it is not possible to define a topology on a set X without taking the whole set and empty set in the collection of open sets.

- This is because the whole set and empty set are required to satisfy the axioms of a topology, namely:

- The union of any collection of open sets is open.

- The intersection of any finite collection of open sets is open.

- The empty set and the whole set are open.

- If we omit the whole set or the empty set from the collection of open sets, then we violate the third axiom and the collection is not a topology.

No, it is not possible. Because then it will violate the 1st axiom of the definition of a topological space.

If you all insist on axiom 1 so strongly, please modify axioms 2 and 3 in the forms that:

2*. tau is closed under non-void unions;

3*. tau is closed under binary (pairwise) intersections.

Otherwise, in contrast to your opinions, one can quite well work with any family tau of subsets of the ground set X.

Prof. Csaszar and his followers, in several papers, assumed only that tau is a weak structure in the sense that the emptyset is in tau.

By using closed sets, the complements of the members of tau, you can

define a generalized closure operation on X in the usual way.

And then, you can use the well established technics of closure spaces.

Generalized closures are much stronger tools than generalized topologies.

In contras to Hausdorff, Sierpinski, Kuratowski and Cech for instance, Bourbaki, Kelley and Engelking had standardized a worse direction by choosing open sets as a starting point.

Though, having in mind the works of Riesz, Efremovic and Smirnov,

they could have choosen some better tools.

Nearness of sets to sets and convergence of nets to nets, are almost as strong tools as families of relations offered by Weil.