Talk:Coherent topology

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[edit] Slight error

It's a good article, but there is a slight error in there I believe. The discrete topology isn't coherent with every collection of subsets. One implication holds of course. If U (as a subset of X) intersected with A for every A in some collection {A} is open in A then U is open i X (because every U is open i X with the discrete topology). The converse doesn't hold. If U is open in X (which it always is) then the intersection of U with A is not necessarily open in every A in {A}.

YohanN7 (talk) 15:56, 2 May 2008 (UTC)

There is no error. Recall that every subset of a discrete space is discrete. So U is open in every A. -- Fropuff (talk) 17:56, 2 May 2008 (UTC)

You can't assume that every subset A of X in a collection of subsets has the subspace topology. That is not part of the definition of a coherent topology (the definition that says shortly U is open in X if and only if U intersected with A is open i A for every A in the collection). The subsets are allowed to have a topology of their own! Suppose that the collection of subsets consists of a single set A with the trivial topology. Let A = {a,b}. Then set set {a} is open in X, but not in A.YohanN7 (talk) 17:50, 5 May 2008 (UTC)

The article states that a discrete space is coherent with every family of subspaces not subsets, so there is no error. As far as I am aware the coherent topologies are defined with respect to subspaces not with respect to arbitrary inclusions. Perhaps I am mistaken. Can you provided a reference for the more general definition? -- Fropuff (talk) 00:20, 6 May 2008 (UTC)

Ok, then what we are discussing is not "right or wrong", but probably merely what one takes for the definition of a coherent topology. My primary reference is actually Willards book referenced in the main article (an excellent book in my oppinion). Choherent topologies are introduced in a problem in chapter 9. I don't have my book at hand at the moment, but i'm fairly sure he uses the word subset instead of subspaces. On the other hand, his convention is to use subset (without further mention of the topology) to mean subspace in the more conventional sense (where the subspace topology is assumed implicitely). So maybe I am the one that is mistaken.

The definition that I (incorrectly) assumed from the start was the one where the subsets are free to have any topology (, and in addition are not required to cover X) does make sense too - it becomes a generalization of the one meant in the article. There are examples: Consider a finite space X with the trivial topology and a collection of subsets which are equal to X as sets, but endowed with some sort of Sierpinski topology. For instance let X = {a,b}, and let one subset have the topology {X,0,{a}}, and another the topology {X,0,{b}}. Then the trivial topology is (the unique) coherent with thos subsets.

Of course, in most typically occuring examples the two definitions coincide. Best regards! YohanN7 (talk) 17:10, 6 May 2008 (UTC)