Fundamental example: The free topological group G over a convergent sequence, in the sense of Graev.

This is the finest topological group with countably many generators, so that the generators converge to the identity. The basic behavior of colimits of Hausdorff compacta plays a crucial role in understanding this construction, sketched as follows.

Appetizer: Fix a group G and show in general that the collection of topologies on G (under which G is a topological group) forms a complete lattice under inclusion, ( key lemma: the union of any collection of topological group topologies on G is a subbasis for a topological group) ( intersections of such topologies behave much more opaquely but save that for dessert).

Main course: Let G be the free group on countably many generators $x_1$, $x_2$,...

Summon the finest topological group topology $\tau$ on G such that $x_1, x_2$,... $\rightarrow$ 1.

The appetizers ensure $\tau$ exists, but is $(G,\tau)$ sequential? Is it metrizable? Is it normal?

The answers are yes, no, and yes, but this is entirely nonobvious. However, much of the mystery is absorbed by basic theory of colimits of Hausdorff compacta.

Notably, $(G,\tau)$ is the direct limit of countably many nested sequential Hausdorff compacta, and hence normal and sequential. Moreover (by an ultimately straightforward miracle), $(G,\tau)$ X $(G,\tau)$ is a sequential space. The utility of the latter fact is that to check continuity of group multiplication, it suffices to check that group multiplication is continuous over convergent sequences.

( To justify the use of `miracle' the plausible claim that the product of two sequential spaces is sequential, is generally false. (Recall a sequential space is a topological quotient of a metric space, or (nonobviously) equivalently, a space whose closed sets are precisely those which are closed under convergent sequences)).

Dessert : Despite the apparent naturality of the construction at hand, we are surrounded by categorical pathology. Notably, fixing the countable free group G and considering the (complete) lattice1 of topological group topologies on G, and the larger (complete) lattice2 of all topologies on G, inclusion of lattice1 into lattice2 is NOT a homomorphism. (The pathology remains even if we restrict the conversation to sequential topologies).

Antacid: The root pathology is created by the general failure of colimits to commute with products. However, fixing a category and making modest adjustments to standard defintions (such as canonically refining the standard product of sequential spaces to preserve the sequential property) can create useful structures and powerful mathematics.