see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
Kolmogorov space, Hausdorff space, regular space, normal space
connected space, locally connected space, contractible space, locally contractible space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
Theorems
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
An equilogical space is a Kolmogorov ($T_0$) topological space $T$ along with an arbitrary equivalence relation $\equiv$ on its points (of note, the equivalence relation need not match the topological structure in any way). A morphism between equilogical spaces $(T, {\equiv})$ and $(U, {\cong})$ is a continuous function $f\colon T \to U$ such that $x \equiv y$ implies $f(x) \cong f(y)$, for all points $x$ and $y$ in $T$. Two such morphisms $f$ and $g$ are considered equal if for all points $x$ in $T$, $f(x) \cong g(x)$.
The category $Equ$ of equilogical spaces obviously contains the category of $T_0$ topological spaces as a full subcategory (by using the trivial equivalence relation of equality on points). Moreover, as opposed to the latter, $Equ$ is in fact cartesian closed; this can be seen using the equivalence of $Equ$ and the category of partial equivalence relations over algebraic lattices.
On the other hand, $Equ$ can be identified with a reflective exponential ideal in the ex/lex completion of the category $Top_0$ of $T_0$ topological spaces. This provides an alternative proof of the cartesian closure of $Equ$, since an exponential ideal in a cartesian closed category is cartesian closed, and $(Top_0)_{ex/lex}$ is cartesian closed (in fact, locally cartesian closed) since $Top_0$ is weakly locally cartesian closed.
Moreover, in this way we can see that the embedding $Top_0 \to Equ$ preserves all existing exponentials, since the embedding $C \to C_{ex/lex}$ does so, and $Equ$ is closed under exponentials in $(Top_0)_{ex/lex}$ and contains the image of $Top_0$. This embedding also preserves all limits, but it does not in general preserve colimits.
The concept was originally introduced for domain theory in a privately circulated manuscript by Dana Scott.
It is then discussed in more detail in
Andrej Bauer, section 4 of The Realizability Approach to Computable Analysis and Topology, PhD thesis CMU (2000) (pdf)
Andrej Bauer, Lars Birkedal, Dana Scott, Equilogical Spaces, 2001 (ps, pdf)