Definition talk:Topology Generated by Synthetic Sub-Basis

I understand your use of $:=$ meaning (I think) "is defined as", but what is $=:$? --Prime.mover 07:34, 26 January 2010 (UTC)

As far as i know, $:$ separates a name from the thing it is defined to refer to, and $:$ is followed by how the name refers to the object. So $a := b$ means that whenever I say $a$, I mean the same object as if I had said $b$. I could also say $a :\Leftrightarrow b$, this would mean that whenever I say $a$, I refer to something that is equivalent to $b$ (whatever that means in the surrounding context). Same goes for things like $f : X \rightarrow Y$ and so on. I'm not sure if this is standard in any way (it's just how I learned it). If there's some established standard around here, just let me know.

And switching it around like $y =: x$ is just the same as saying $x := y$. Sometimes it's more readable: Whereas $x := y$ is like "I will use the name $x$ for the following concept ($y$)", $y =: x$ is more like "Thus we arrive at the following concept ($y$), which I'll call $x$". But it's a matter of personal taste. Again, if there's already a standard on how things are done, let me know. If not it might be a good idea to think about such a standard. --Florian Brucker 11:35, 29 January 2010 (UTC)

Equivalence of Definitions

Are these definitions equivalent, or do they produce different results? (I haven't taken the time to analyse the implications so I don't know.) If the first, we need a page proving that equivalence - if the latter we need to separate the definitions out and disambiguate them somehow, or we are going to tie ourselves in knots when linking to this page. --prime mover (talk) 08:34, 23 September 2012 (UTC)

I suspect equivalence, and I suspect a non-trivial exercise is to prove this. I don't care much for this general topology stuff but I think one can proceed by showing that the topology from def. 1 satisfies def 2. --Lord_Farin (talk) 09:41, 24 September 2012 (UTC)