|WikiProject Mathematics||(Rated Start-class, Mid-importance)|
To: User 184.108.40.206
Probably you forgot to log in, so I address you by your IP address.
I will answer your questions, and add some comments.
Moving from most concrete to most general...why omit metric spaces??
- Good point. Because I was trying to make things clear, without being most general. But your solution, to first put an intuitive explanation, is better.
also "x belonging to S" is not required in def..
- Another good point. Again, I thought that was not wrong, and more clear that way. But I agree with you, the way you put it now is better.
also, int[1,10] = (1,10)
- I don't understand that. This is how it was before too. Was there any mistake?
- There was no mistake. I was emphasising that this relation only holds for the Euclidean topology. In the original, it simply says, "int[1, 10] = (1, 10)". Strictly speaking, this statement has no meaning until you specify what the topology is. Of course, everyone assumes its Euclidean topology, but this should still be said explicitly. One of the most common errors beginners make is assuming that the usual Euclidean topology is the only possible topology for R. Revolver 08:17, 11 Dec 2004 (UTC)
- Got it! That statement, "int[1, 10] = (1, 10)", was put there before me. I agree with you, after the person reading that page understands what topology is about, then one has to indeed point out that given a set, there can be many topologies on it, and then the notion of opnenness might not be the intuitive thing we are used to from Euclidean spaces. --Olegalexandrov 19:13, 11 Dec 2004 (UTC)
One more thing. Could you make a page called Lower-limit topology. You link to it, but this is not of much help if the page itself is not available. (OK, I know the Wikipedia principle, somebody else sooner or later will write that page, but still, a better solution is to actually write that page.)
- Sorry...in honesty, I thought the page already existed. Maybe I spelled it wrong. Revolver 08:17, 11 Dec 2004 (UTC)
- Awesome! --Olegalexandrov 19:13, 11 Dec 2004 (UTC)
And lastly, the Interior (topology) page was in a bad shape before I changed it, and you made it much better. Thanks! --Olegalexandrov 18:31, 10 Dec 2004 (UTC)
- Thanks. I'm looking to make some similarity to the closure page. That page seems a little abstract and confusing at first. Revolver 08:17, 11 Dec 2004 (UTC)
- I agree with you! I had also noticed the closure page is quite confusing. --Olegalexandrov 19:13, 11 Dec 2004 (UTC)
I found the definitions going from the most specific to the most general a little disorienting. I remember learning it the other way round. Is this in fact now the preferred way? Martin Packer (talk) 20:56, 9 April 2008 (UTC)
- I think it is an attempt to be friendly to readers who haven't studied any topology. It may indeed be better to separate this into an informal exposition that is not under the heading of "definition", and a real definition that applies to all topological spaces, the easiest of which is probably the largest open subset. --Lambiam 17:18, 10 April 2008 (UTC)
I added a definition that says what interior points of S are: points contained in an open subset of S. I like this definition because "open set" can be taken as the fundamental notion for topology, and often is, with other things defined in terms of it. Granted the same could probably be done for various other concepts used in the other definitions, but as things stand I find the other definitions in the lead section a bit distracting, as they require chasing through several other pages to unfold. MorphismOfDoom (talk) 02:51, 23 August 2014 (UTC)