Instant topological relationships hidden in the reality

Abstract

In most applications of general topology, topology usually is not the first, 
primary structure, but the information which finally leads to the construction of
the certain, for some purpose required topology, is filtered by more or less 
thick filter of the other mathematical structures. This fact has two main 
consequences:

(1) Most important applied constructions may be done in the primary 
structure, bypassing the topology.

(2) Some topologically important information from the reality may be lost 
(filtered out by the other, front-end mathematical structures).

Thus some natural and direct connection between topology and the reality 
could be useful. In this contribution we will discuss a pointless topological 
structure which directly reflects relationship between various locations which 
are glued together by possible presence of a physical object or a virtual 
``observer".