[Topology] why the words finer and coarser ?

[nonequilibrium]

[Topology] why the words "finer" and "coarser"?

Hello,

I'm following an introductory course on topology.

Why is it that a topology with lots of opens is called fine, and one with a few ones is called coarse?

More specifically: why is this terminology more logical than the reverse (i.e. calling one with a small number of open sets fine)? (so to be able to remember it better; it seems to arbitrary, but maybe it isn't)

 

[quasar987]

I like to think of the open sets as forming a "granulation" of the space. In a space with few open sets, the gradulation is going to be very coarse, while in one with lots of open sets, it will be finer.. This is how I remember the meaning of the terms.

 

[HallsofIvy]

It's the size of the "open sets", in the same sense that a grind of flour is "finer" than another if it has been ground to a smaller size. Topology A is "finer" than topology B if and only if its contains all the sets in B and contains some additional ones that are subsets of sets in B. And, of course, in that case, B is "coarser" than A.

Given any set, X, the "discrete topology", where every subset of X is open, is "finer" than any other topology on X while the "indiscrete topology",where only X and the empty set are open, is "courser" than any other topology on X.

 

[nonequilibrium]

But is there a reason to not use the term finer for a lot of closed sets, instead of a lot of open sets? (if we were to redefine topologies in terms of closed sets instead of open sets)

Or is this just a convention?

 

[Jamma]

I think that it can actually be used the other way round, which really confuses things! (it will be pointed out in the text though). But usually finer does mean more open sets, and as others have pointed out, this is because we think can think of the space as being a lot finer, with points "more separated".

But is there a reason to not use the term finer for a lot of closed sets, instead of a lot of open sets? (if we were to redefine topologies in terms of closed sets instead of open sets)

Or is this just a convention?

But for each open set there is a closed set (it's complement) so things shouldn't be any different. Topologies are usually defined with open sets, so it's just not an issue that comes up, and wouldn't effect things if it did (a closed set in one topology is in another topology if and only if its complementary open set is also an open set in the other topology, which would be so for a finer topology).

I actually prefer the terms "stronger" and "weaker". Stronger means larger so that, if you have an open set in one topology, then you know that a "stronger" one will have it also. It's all just taste really.

 

[nonequilibrium]

But for each open set there is a closed set (it's complement) so things shouldn't be any different. Topologies are usually defined with open sets, so it's just not an issue that comes up, and wouldn't effect things if it did (a closed set in one topology is in another topology if and only if its complementary open set is also an open set in the other topology, which would be so for a finer topology).

True! I should have thought that one through more carefully.

And thanks for your reply :)

 

[Deveno]

the way i always remember it is to think of the extremes:

the indiscrete topology just has one non-empty open set, the entire space X.

from a topological point of view, we just have one big "blob" that we can't see into.

the discrete topology has every subset of X, as an open set, so in particular singleton sets {x} are open.

this is like every point being a space unto itself, so that the topological structure on X is very fine like a bag of sugar.

the standard topology in euclidean space (usually given by a metric based on euclidean distance), is rather fine, there are plenty of open sets to be had, which makes it "flexible".