https://huji.zoom.us/j/84395259634?pwd=aNXbRjiAA1mae1ddbJubRXEZ3mrbat.1 

 
Title: Properties of continuous maps whose complete definitions fit into a single byte
Abstract: We consider a binary relation on continuous maps, called the lifting property, used prominently by Quillen to define an axiomatisation of homotopy theory. 
 
What can we define just by taking several times left and right orthogonal complements with respect to this binary relation starting from the class $\{ \emptyset \to \{*\} \}$consisting of a single map from the empty set to a singleton ?
 
It turn out in this way we can define precisely 21 different classes (=properties) of continuous maps, and 9 of them are in fact introduced explicitly in a typical first year course of topology: 
having a section, dense image;
subsets, closed subsets (that is, immersions and closed immersions); 
quotients; 
surjective, injective; 
the domain is empty or non-empty; 
homeomorphism.
 
We can also define properties of spaces being connected, having a generic point, satisfying Separation Axiom $T_0$ and $T_1$. 
As we need to take the orthogonal complement less than 8 times, each definition fits into a byte. If we start with a class $ \{ f \} $ where $f$ is a map of spaces of size at most 5, then we can also define properties of a space being compact, contractible, connected, totally or extremally disconnected, (hereditary) normal, zero-dimensional, discrete, antidiscrete. 
We can also define notions such as induced topology, bijections, open subsets, and a few others.
This explicates finite combinatorics implicit in these basic definitions of topology.
Taking left and right orthogonal complements defines an action of the free semigroup with two generators; we draw the Schreier graph of the orbit of $ \{ \emptyset \to \{*\} \} $.
 
This is joint work with A.Lavan.