NetNews Usenet Archive 1992 #26/NN_1992_26.iso/spool/sci/math/14491

Text File | 1992-11-07 | 1.7 KB | 44 lines

Newsgroups: sci.math
Path: sparky!uunet!stanford.edu!leland.Stanford.EDU!leland.Stanford.EDU!ledwards
From: [e-mail] (Laurence James Edwards)
Subject: Re: definition of topological space
Message-ID: <[e-mail]>
Keywords: Topology; Open sets; Continuity
Sender: [e-mail] (Mr News)
Organization: DSG, Stanford University, CA 94305, USA
References: <[e-mail]> <[e-mail]>
Date: Fri, 6 Nov 92 09:20:37 GMT
Lines: 31

In article <[e-mail]>, [e-mail] (Richard Pinch) writes:
|> In article <[e-mail]>
|> [e-mail] (Laurence James Edwards) writes:
|> [...]
|> Well, here goes.  The sets in the family are called "open" and you should
|> think of them as being rather like the open intervals in the real line.
|> A subset of R is open iff it is the union of a collection of open intervals
|> (a,b); equivalently if it contains an open interval round any of its points.
|>
|> The epsilon-delta definition of continuity says that f is continuous iff
|> for all x, for all e > 0, there exists d > 0 such that
|>      |x-x'| < e => |f(x) - f(x')| < d
|> for all x, for all e > 0, there exists d > 0 such that
|>      f*(f(x)-d, f(x)+d) contains (x-e,x+e)
|> where f*(Y) is the set of x such that f(c) is in Y.
|> i.e.
|> Y open => f*(Y) open
|>
|>  and this last is the topological definition of continuity.

In defining continuity what is the advantage of using open sets vs. closed sets,
i.e. why not say:

 |x-x'| <= e => |f(x) - f(x')| <= d

and similarly:

 Y closed => f*(Y) closed


Larry Edwards