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1 - Can someone help understand what they mean by the following Topological Space:

Let T be the family of subsets of R (the reals) defined by: A subset K or R belongs to T if and only if...

...for each r in K there are real numbers a, b such that...

a < r < b and {x: xeR, a < x < b} C K.

Note: C was used to show that it is a subset of K.

Are they trying to say that K is finite?

2 - For the Discrete Topology for X, which subsets are closed?

My answer is that they are all closed and open because all the subsets are included, hence they are all open, but by definition the complement of each of those sets is closed. Since the complement the those sets are also possible subsets of X, they are all open since the Discrete Topology contains all the subsets.

3 - For the Indiscrete Topology for X, which subsets are closed?

My answer is all the subsets of X are closed. The Indiscrete Topology for X only contains the void set and the underlying set X, which both of which are also closed, as well as every subset of X.

I'd appreciate any comments.

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# My Topology Questions

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