Give an example of a closed pointset and a continuous , such that the image is not closed.
I can not seem to come up with a suitable example. I need help on this problem. In our book, pointsets are subsets of Baire space. Also, denotes Baire space. Thanks in advance.
Hmm, well here's maybe an easier one. being a complete metric space is a Baire space, right? Now, Remembering use that and the fact that the canonical projection x,y)\mapsto x" alt="\pi:\mathbb{R}^2\to\mathbb{R}x,y)\mapsto x" /> is continuous but is closed but . So, I think you can go from there.
I think this is a common misunderstanding.
Drexel28 is speaking about Baire space as a topological property. (Roughly speaking, it is a topological space in which Baire category theorem holds.)
The original question of eskimo343 concerns the Baire space, which is a topological space on the set of all finite sequences of integers with the topology given in some sense by the tree structure. (It is homeomorphic to irrationals.)
BTW the same question was also posted here: example, closed pointset
and here: S.O.S. Mathematics CyberBoard :: View topic - example, closed pointset
and here: http://www.mathlinks.ro/viewtopic.php?p=1849578#1849578
I'll use the terminology I know from descriptive set theory, it should probably be almost the same as in Moschovakis' book Notes on set theory. (The OP is probably using this book, since the question appears as one of the problems in this book.)
Let L be the sets of all strings of length 1 with the exception of (0). I define as follows:
, if and
, otherwise.
Then is not a body of a tree, hence it is not closed. (Proposition 10.7)
The map f is continuous since:
is open;
is open;
is open for .
(I have used the definition of continuity using basic neighborhoods, perhaps another approach from Theorem 10.15 is also possible - I did not try that.)