Hi there, I've got quite a big assignment about topological spaces and think I'm doing OK apart from this bit...
Let X and Y be topological spaces. Show that is continuous if and only if
. By considering the map , show that we
do not expect .
For the first part I know, f is continuous if and only if the image of the closure of every subset is contained in the closure of the image which implies the result but don't think this is really me showing this, its just me stating a rule...
For second part I'd have thought we would expect so I'm obviously not understanding something properly.
Any help much appreciated!!
Well for the first part I'm basically just quoting out of the lecture notes we have been going over. I 'know' it because it's given as a rule but guess they're looking for me to actually prove it which I'm not sure on how to do! For the second part I can see it would make sense if that were the question but in both hand out and online the question we have is one stated. Worse comes to worse I'll just solve it for and just see what happens!
Many good topology textbooks, such as Helen Cullen's, give a discussion of alternate definitions of continuity . It appears that you are being asked to show that the statement is an alternate definition of continuity. What is you basic definition?
As for the function , it does not have the suggested property but does.
Recalling that
a closed set.
But , not closed.
Yeah can see that second part now, thanks!!
The definition for continuous we have is...
Let and be topological spaces. A function
is continuous iff
implies .
That is, inverse images of open sets are open.
It even says lower down that the part they're asking about is just an alternate definition. I'm guessing you have to do some playing around with complements and things as thats what most of the stuff so far has been like but don't really know. Sorry, bit unclear but I don't really know what it wants myself!