Science Forums 2

[Qfwfq]

Originally Posted by sanctus

This is obvious if S is an empty set, but I was wondering if S contains say 2 points then you could say that S contains at leas 4 subsets, the ones above plus the ones including each point only.

Actually, the odd-man-out is because in this case coincides with it. If cardinality is 1 then and are the only ones, but distinct. For greater cardinality of course there are all the others.

Originally Posted by sanctus

Can't you also elaboarte it further then and say that every set S has 2+n! subsets where n is the number of points in S? I.e. 2 from and and then the possible order independent combinations of the elements/points in S...

I always thought it's more like , and with this count including and . Each element is either in or out; if they're all in it's the whole of , if they're all out it's .

Originally Posted by maddog

I would like to understand better the significance of a Topological Space.

OK, for the moment we could be precognitive and say that it's an abstraction of some things which are definable in terms of a metric. Actually it's the notion of "open set" that can be so defined and then shown to have those properties. The metric is therefore said to induce a topology, called the topology of the metric.

Ultimately, it all leads to there being some aspects of some things that are topological rather than metric ones, in the sense that they don't change when the critter is "deformed with continuity", but of course this has to yet be defined here.

Originally Posted by maddog

Also, what it means when a said space if "compact" or as in "compact cover". This is of
course presuming we have some kind of metric and a Hausdorf space definition where
conitinuity is in force. Thus we have neighborhoods about each member of the space
such that for any neighborhood of a member of
contains another member of .

Gosh give the bloke a chance to get that far!