For some "structures" (in informal sense for a lack of a formal term) in mathematics, such as groups, rings, and vector spaces, a bijective homomorphism is an isomorphism; i.e. the inverse is also a homomorphism. For some other structures, such as topological spaces and differentiable manifolds, a bijective homomorphism may not be an isomorphism.
Are there characterizations of sub-classes of structures which have the property that a bijective homomorphism is an isomorphism? For example, do all algebraic structures (in formal sense this time) have this property?
Recall that structures and their structure preserving maps often assemble themselves into categories. So, there is a category $Grp$ of groups, $Ab$ of abelian groups, $Ring$ of rings and so on. Now, if the structures are based on sets, then often there will be a forgetful functor $C\to Set$ from the category $C$ to the category of sets and functions. The property you are looking at is reflection of isomorphisms by this functor. So, one can ask, for a given category $C$, when is the forgetful functor $C\to Set$, assuming it exists, reflects isomorphism? A pretty far reaching answer is that whenever $C\to Set$ is monadic. Now, that latter term is a bit more technical, but, in a nutshell, $C\to Set$ is monadic if $C$ is a category of nice enough algebraic structures. Monadicity captures many algebraic structures, but not, for instance, posets (if you consider these algebraic): the forgetful functor does not reflect isomorphisms.
Interestingly, this notion of reflection of isomorphisms is in fact one of the conditions of Beck's Monadicity Theorem characterising monadic adjunctions.