CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The Tychonoff product (named for Andrey Tikhonov) is simply the product in the category Top of topological spaces and continuous maps.
It is the usual notion of ‘product’ of topological spaces, but should be distinguished from the box product?, which is sometimes useful. (They are the same for finite products.)
Let $(X_i)_i$ be a family of topological spaces. Consider the cartesian product $\prod_i {|X_i|}$ of the underlying sets of these
The Tychonoff topology or product topology on $\prod_i {|X_i|}$ is the initial topology generated by the $\pi_i$. The Tychonoff product or topological product or simply product of the spaces $X_i$ is the set $\prod_i {|X_i|}$ equipped with the Tychonoff product topology.
Of course, the maps $\pi_i$ are continuous maps, so we have a cone in Top.
This is in fact a product cone in $Top$.
By the general characterization of limits in Top, see here.
The Tychonoff topology on $\prod_{i \in I} X_i$ is equivalently the topology
generated from the sub-basis
generated from the basis
where $p_i \colon \prod_{j \in I} X_j \to X_i$ are the defining projection maps.
In particular if $I$ itself is already finite, then a basis of the product topology is given by the Cartesian products of elements of a basis in the factors.
Terence Tao, Notes on products of topological spaces (pdf)
Florian Herzig, Product topology (pdf)