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Represented spaces form the general setting for the study of computabilityderived from Turing machines. As such, they are the basic entities forendeavors such as computable analysis or computable measure theory. The theoryof represented spaces is well-known to exhibit a strong topological flavour. Wepresent an abstract and very succinct introduction to the field; drawingheavily on prior work by Escard\'o, Schr\"oder, and others.  Central aspects of the theory are function spaces and various spaces ofsubsets derived from other represented spaces, and -- closely linked to these-- properties of represented spaces such as compactness, overtness andseparation principles. Both the derived spaces and the properties areintroduced by demanding the computability of certain mappings, and it isdemonstrated that typically various interesting mappings induce the sameproperty.

On the topological aspects of the theory of represented spaces