defined by
is this function continuous, and is this function closed
I found it closed and not continuous
am I right ?
Thanks in advance
What are and ? The usual and cofinite topologies? And I assume .
I agree that it's closed. If then and so is open. The image of any subset of is closed.
I also agree that it can't be continuous. For, if were continuous then where is the subspace topology would be continuous. But, every subspace of a cofinite space has the cofinite topology, and a finite cofinite space is discrete. And so, where is the two point discrete space.
Thus, if were continuous there would be a map which is surjective and continuous. What's the problem with that?
Yeah it is the usual and cofinite topologies and the set is the real numbers
but I do not know how to write R with two lines
about your question
we will face a contradiction
suppose D= {1,2}
and open sets in
and
that means real numbers with usual topology is connected contradiction