Q:

The category of Compact Hausdorff spaces is special: why? In which other contexts bijections are automatically isomorphisms of objects?

I am writing my bachelor thesis, mainly about General Topology and Topological Vector Spaces. Moreover I know a little bit about Category Theory: categories, functors, natural transformations, representability and the Yoneda Lemma. 
A simple consideraation is the following:  
Any continuous function between a compact and an Hausdorff space is closed  
As an immediate consequence a continuous bijection between two compact Husdorff spaces is automatically a homemorphism.
This motivates two facts: 

Adding just an open set the topology ceases to be compact and
removing one the topology cease to be Hausdorff.  Hence the topology of a CHaus space is 'final' with respect to the property of compactness and 'initial' with respect to that of Hausdorffness
A bijective
morphism in the category CHaus is automatically an isomorphism

Now my questions are:

First of all: are 1 and 2 'categorically' related?
Secondly: I think a completely analogous result is interpreting the Banach isomorphism theorem in the category Ban of banach spaces. What's underlying? What do these category share? Can we generalize? Do we have more examples, specially in Topology/Functional Analysis?
Third: can someone suggest some nice 'easy' application of category theory to general topology or functional analysis? I mainly saw algebraic and algebraic topological ones.

Thanks in advance

A:

This is more of a long comment than an answer. We call a category $C$ concrete if it's equipped with a forgetful functor $U : C \to \text{Set}$, usually assumed to be faithful; this formalizes the intuitive notion of a category of "sets with extra structure," where $F$ describes the underlying set of an object. The property you want, that a morphism in $C$ which is bijective on underlying sets is an isomorphism, corresponds to $U$ being conservative. A conservative functor is one that reflects isomorphisms, meaning that if $F(f)$ is an isomorphism then $f$ is an isomorphism.
Faithful and conservative functors can be related as follows. First, some nonstandard definitions: say that a morphism is a pseudo-isomorphism if it is both a monomorphism and an epimorphism, and a fake isomorphism if it's a pseudo-isomorphism, but not an isomorphism.

Exercise 1a: Faithful functors reflect epimorphisms and monomorphisms: that is, if $F$ is faithful and $f$ is a morphism, then if $F(f)$ is an epimorphism then $f$ is an epimorphism, and if $F(f)$ is a monomorphism then $f$ is a monomorphism. Hence faithful functors reflect pseudo-isomorphisms.
Exercise 1b: If $F : C \to D$ is a faithful functor and $C$ has no fake isomorphisms (so every pseudo-isomorphism is an isomorphism), then $F$ is conservative.

Hence, if $C$ is a concrete category whose forgetful functor isn't conservative, then $C$ must have fake isomorphisms. $C = \text{Top}$ is a well-known example; in this category fake isomorphisms exist because we can add open sets to a topology and get another topology, which allows us to construct continuous bijections which are not homeomorphisms.
In addition, while it's not true in general that pseudo-isomorphisms are isomorphisms, there are many statements of the form "a morphism which is both a monomorphism and a (some special kind of epimorphism) is an isomorphism." A reasonably useful one in practice is:

Exercise 2a: A morphism which is both a monomorphism and an effective epimorphism is an isomorphism.
Exercise 2b: If $F : C \to D$ is a faithful functor and every epimorphism in $C$ is effective, then $F$ is conservative.

The condition that every epimorphism is effective holds in some categories of algebraic objects, such as $\text{Vect}$ and $\text{Grp}$, but not in others, such as $\text{Ring}$.
It turns out that in $\text{CHaus}$ every epimorphism is effective; what this says somewhat more concretely is that every continuous surjection $X \to Y$ between compact Hausdorff spaces is a quotient map, or in other words that $Y$ has the quotient topology (note that this is emphatically not true in $\text{Top}$!). so this is one way of explaining why $\text{CHaus}$ has a conservative forgetful functor. I don't think this is true in the category of Banach spaces though.
The comments allude to the fact that monadic functors are conservative, and while this covers the case of compact Hausdorff spaces it doesn't cover the case of Banach spaces.

