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Math Help - clopen sets in Std Topology

  1. #1
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    clopen sets in Std Topology

    Which sets are closed and open in R^2 with the Std topologgy?

    I am confused about this, help plz!! THis is a question that was made while looking at boundaries. Can you make a general statement?
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    Which sets are closed and open in R^2 with the Std topologgy?

    I am confused about this, help plz!! THis is a question that was made while looking at boundaries. Can you make a general statement?
    The sets \emptyset, \mathbb{R}^2 are both open and closed.
    Try it out, what is the definition of being open and closed?
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  3. #3
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    Another example can be found in a disconnected topological space.

    A topological space X is disconnected iff X has a proper subset A which is both open and closed.

    For instance, the subspace X=(0,1) \cup (2,3) of R is a disconnected space.
    Both (0,1) and (2,3) are open in X.

    Definition: A set C is closed in X if X\C is open.

    Now, a complement of (0,1) in X is (2,3) which is open in X. Thus, (0,1) is a closed set in X. So, we can conclude (0,1) is a clopen set in X.

    You can check that a boundary of (0,1) in X is empty as well (same with (2,3)).
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