This quote is from the wikipedia page on topoi:
Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic.
What does this article mean by "topological heuristics"? This suggests to me that there are certain principles that are in a sense "deeper" than topology, that have application in broader contexts. What are those principles?