Why are topologies typically defined with open sets rather then closed sets? In general one can define a topological space either in terms of its open sets or in terms of its closed sets, however it seems that depending on the context the "closed sets" might make more intuitive sense, for example suppose that some class of geometric structures is closed under isomorphisms with respect to arbitrary intersections and finite unions. Now if we look at these as the open sets of a topology on some specific sets by taking complements it may be that this isomorphism between the structures no longer works as the complements are relative with respect to the points, thus it would make more sense to study these as closed sets rather then complementing them and looking at them as open sets.
Of course though it is simpler to adopt one of the two representations when defining a bunch of associated terminology otherwise you'll end up having to do this twice, with roughly the same definition only you're running around complementing every set in the corresponding definitions. Is that what this is just an arbitrary convention to save space i.e. people just use a single notion to save time?
 A: The history of topology is far from being a long quiet river, and the definition of a topology using open sets emerged after a long maturation process. I strongly recommend you Moore's fascinating survey [3] for a complete account of the story, in particular Section 14. Here are two quotes from this section which should answer your question:

As the group of French mathematicians collectively known as Nicolas
Bourbaki was deciding how to treat general topology in the years
1935–1938, they began with a mix of concepts taken from Fréchet,
Riesz, Weyl, Hausdorff, and Aleksandrov. (...) On André Weil's
suggestion, Bourbaki used the concept of open set as the sole
primitive idea. In the first published edition [1, 1940] of his chapter
“Structures topologiques” Bourbaki used the concept of open set as the
sole primitive idea, and as sole axioms a slight variant on the first
of those that Aleksandrov had used in 1925: the intersection of any
finite set of open sets is open, and the union of any set of open sets
is open.


In the United States the most influential topology textbook for
several decades (beginning in 1955) was undoubtedly John L. Kelley’s
General Topology [2]. (...) Kelley was familiar with Bourbaki’s work and
adopted precisely Bourbaki’s two axioms for open sets. Later
treatments of topology would usually spell out explicitly the two
further axioms that the empty set and the whole space are open. These
four axioms for a topological space, expressed using open sets alone,
then became standard. Many textbooks on general topology appeared in
the later decades of the twentieth century, and they all used these
same four axioms. As far as general topology was concerned, the
competition for which concept was most fundamental had ended with the
modern definition of a topological space based on open sets.

[1] Bourbaki, N., 1940. Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques. Actualités scientifiques et industrielles, vol. 858.
Hermann, Paris.
[2] Kelley, J.L., 1955. General Topology. Van Nostrand, Princeton, NJ.
[3] Moore, Gregory H. The emergence of open sets, closed sets, and limit points in analysis and topology. Historia Math. 35 (2008), no. 3, 220--241.
A: The first 30 or so pages of Topology and Groupoids by Ronnie Brown, particularly the early part of Chapter 2, give some insight into this issue. There are a number systems we can use for topological spaces, including:

*

*the neighborhood axioms

*the open set axioms

*the closed set axioms

*the closure operator

*the interior operator

*the relation $A \subseteq \text{Int} B$.

These are all, as far as I know, equivalent. I sometimes think of it like a road intersection---they are different ways of arriving at the same place. I am hardly a topologist but the message that I've internalized is to use whichever perspective is most helpful for the matter at hand.
Further reading:

*

*Topology and Groupoids

*Kuratowski closure axioms

*Professional mathematicians discussing something similar at MathOverflow.

