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I'm looking at some examples in the Topology: Pure and Applied text.

Looking at example 2.1 Consider A=[0,1) as a subset of R with the standard topology. Then Aint=(0,1) and Aclos=[0,1].

Can someone explain to me why the union of all open sets in A is that?

Furthermore, example 2.3

Consider A=[0,1) as a subset of R in the finite complement topology. Here Aint = 0 because there are no nonempty open sets contained in [0,1). Since A is infinite, and the only infinite closed set in this topology is R, it follows that Aclos=R.

I have the same question as above, but also,why is A infinite? And more importantly, the book described R as being open when proving the the finite complement is a topology, and now they are saying its closed?

Please help!!