defined by

is this function continuous, and is this function closed

I found it closed and not continuous

am I right ?

Thanks in advance

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- April 27th 2010, 12:32 PMAmerContinous function in topologies

defined by

is this function continuous, and is this function closed

I found it closed and not continuous

am I right ?

Thanks in advance - April 27th 2010, 12:52 PMDrexel28
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? - April 27th 2010, 01:05 PMAmer
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 - April 27th 2010, 01:08 PMDrexel28