Why are "the" morphisms of the category of topological spaces continuous maps? On the Wikipedia page for "morphism (category theory)" it says that:

In the category of topological spaces, morphisms are continuous functions and isomorphisms are called homeomorphisms.

In what sense would this make sense? Is there some abstract or technical reason why "the" morphisms "are" continuous functions, rather than something else, like functions that map open sets into open sets? 
 A: It's a bit like asking "Why in the set $\mathbb{N}$ the elements are $0$, $1$, $2$...?" The answer is: because we define $\mathbb{N}$ like that. If we chose different elements it would be an equally "valid" set, but it would be a different set and we wouldn't call it $\mathbb{N}$.
For many applications, we want to consider the category with objects topological spaces and morphisms continuous functions. And it's so prevalent that people decided to call this category "the category of topological spaces" without more qualifiers, simply because it's shorter than "the category of topological spaces and continuous functions". If you manage to cook up a category where the objects are topological spaces but the morphisms are something else (maps that aren't necessarily continuous, open maps...), then it's a "valid" category too, but it's not the one that most people call "the category of topological spaces".
A: If open maps (functions that map open sets to open sets) are taken as the morphisms, then there are very few continuous (in the usual sense) functions that belong to the category.   There are quotient maps onto open subsets of the target space, but no embeddings of a space into one with "thicker" open sets. 
Some of the things this excludes are: parametric curves $[0,1] \to \mathbb{R}^2$, constant functions on any nontrivial space, the diagonal map $Y \to Y \times Y$, and embeddings $A \to A \times B$.    
Maybe if you restrict the discontinuities of the open maps there are some interesting jigsaw puzzle-like functions in the category but basically the only things that have a connection to the usual topological category are (generalized) coverings of open sets.
