Comparison of Topologies
A variety of topologies can be placed on a set to form a topological space. When every set in a topology τ1 is also in a topology τ2, we say that τ2 is finer than τ1, and τ1 is coarser than τ2. A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in the literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading.
The collection of all topologies on a given fixed set X forms a complete lattice: if F = {τα| α in A} is a collection of topologies on X, then the meet of F is the intersection of F, and the join of F is the meet of the collection of all topologies on X that contain every member of F.
Read more about this topic: Topological Space
Famous quotes containing the words comparison of and/or comparison:
“We teach boys to be such men as we are. We do not teach them to aspire to be all they can. We do not give them a training as if we believed in their noble nature. We scarce educate their bodies. We do not train the eye and the hand. We exercise their understandings to the apprehension and comparison of some facts, to a skill in numbers, in words; we aim to make accountants, attorneys, engineers; but not to make able, earnest, great- hearted men.”
—Ralph Waldo Emerson (1803–1882)
“I have travelled a good deal in Concord; and everywhere, in shops, and offices, and fields, the inhabitants have appeared to me to be doing penance in a thousand remarkable ways.... The twelve labors of Hercules were trifling in comparison with those which my neighbors have undertaken; for they were only twelve, and had an end; but I could never see that these men slew or captured any monster or finished any labor.”
—Henry David Thoreau (1817–1862)