Open/Closed continuous maps between the plane

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Homework Statement


Give examples of maps between subsets of the plane (with Euclidean toplogy) that are:
a) open but not closed or continuous
b) closed but not open or continuous
c) continuous and open but not closed
e) continuous and closed but not open
f) open and closed but not continuous

Homework Equations




The Attempt at a Solution


So i just want to get this thread up now and then update it as I work on each of these individually.... I have a few thoughts so far

e) How about a map R^2-->R^2 that sends each basis element to it's closure?
c) Maybe a map R^2--->R^2 that sends the closure of each basis element to it's interior?

I will think about this more and post more ideas for the other ones... If anyone could critique what I have thought of so far I'd appreciate it :D.

Thanks PF!!
 

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  • #2
andrewkirk
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Do we need one more to complete the set of (a)-(f): A continuous map that is not open or closed?

Also, I think it is important to be explicit about what we mean by open and closed. Literally, open (closed) means that it maps sets that are open (closed) in the topology of the domain to sets that are open (closed) in the topology of the range. Note that those are both subspace topologies. If the domain is a proper subset of the plane, a set may be open (closed) in the domain but not in the plane. The same applies to the range.

One consequence of this is that in some cases we may be able to convert a map from not-open (not-closed) to open (closed) simply by changing the range.
 
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