I am trying to understand how the usual topology on $R^n$ fits into the lattice of topologies. In particular, I am wondering what is the greatest lower bound and least upper bound.
I have been asked to clarify the question. Thus, (borrowing in considerable part from Joel David Hamkins) I will expand it as follows: This question is about the position of the usual topology in the space of all topologies on a fixed set X. We may order the topologies by refinement, so that τ ≤ σ just in case every τ open set is open in σ. Equivalently, we say in this case that τ is coarser than σ, that σ is finer than τ or that σ refines τ. (See https://en.wikipedia.org/wiki/Comparison_of_topologies.) The least element in this order is the indiscrete topology and the largest topology is the discrete topology.
One can show that the collection of all topologies on a fixed set is a complete lattice. In the downward direction, for example, the intersection of any collection of topologies on X remains a topology on X, and this intersection is the largest topology contained in them all. Similarly, the union of any number of topologies generates a smallest topology containing all of them (by closing under finite intersections and arbitrary unions). Thus, the collection of all topologies on X is a complete lattice.
For example the compact topologies are closed downward in this lattice, since if a topology τ has fewer open sets than σ and σ is compact, then τ is compact, and the Hausdorff topologies are closed upward, since if τ is Hausdorff and contained in σ, then σ is Hausdorff. Thus, the compact topologies inhabit the bottom of the lattice and the Hausdorff topologies the top.
The lattice has atoms, since we can form the almost-indiscrete topology having just one nontrivial open set (and any nontrivial subset will do). It follows that every topology is the least upper bound of the atoms below it.
My question then is, considering the usual topology on $R^n$, what is the GLB and the LUB of this topology in the lattice? Or, if this is not known, what topologies are known to be comparable to it in the upward and downward directions?
For n = 1, the lower and upper limit topologies and the add an additional open set topology are finer than the usual R.
A topology of a partition of R is coarser provided at least one partition part is multipoint. Other coarser spaces are the open and closed ray topologies, the cofinite and cocountable topologies. A coarser, uncountable, Hausdorff space eludes me.