When I read a proof in the Explorer I find it difficult to remember which of the two symbols ( Cls or cls ) means closure and which means closed . In my opinion, we should use clearer symbols (clu and cls for instance). – fl 9-Jun-2007
I believe that both symbols are used in the literature (although not always both in the same book). However, there is an "easy" way to remember these, as well as others. An object starting with a capital letter usually denotes a collection that can be interpreted as a property, in the sense that A e. Obj means "A is an Obj". An object starting with a small letter usually denotes a function. Now both Cls and cls are "functions" in the sense that they depend on the topology J, but in their final use, they conform to this standard. Thus ( Cls ` J ) is the set of all closed sets and A e. ( Cls ` J ) means "A is a closed set". On the other hand, ( cls ` J ) takes in a set and returns an image of it, i.e. its closure. Thus ( ( cls ` J ) ` A ) returns the closure of A.
I mention the above because this general convention of capital and small letters is used in several places and can help you recognize things easier. Of course there are exceptions you can find - I try to do whatever the literature does when there is a standard symbol - but when there is a choice, this is the convention that I try to follow. An easy way to remember the convention is to think of sin, cos, log, etc. which are functions and always begin with a small letter. On the other hand, things like RR and CC - collections - are often capital R and C in books, and it would be odd to see "r" and "c" for the sets of real and complex numbers. "A e. RR" means "A is a real number". Similarly, books use Top (not "top") for the collection of all topologies.
That being said, I will think about changing "Cls" to "Clsd" for clarity. – norm 9 Jun 2007
Thank you Norm. – fl