Topology Question: Is A Open if Open Subsets of A Union to A?

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The discussion centers on the question of whether a subset A of R can be concluded to be open if it is represented as a union of open subsets of A. The consensus is that A cannot be deemed open in R solely based on the fact that it is open in itself. A counterexample provided illustrates that a non-open subset A can still be open when considered as a subset of itself, thus confirming that the openness of A in the context of R remains indeterminate.

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JasonJo
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Suppose I have some subset of R, not necessarily an interval, let it be denoted as A. I have some union (might be countable, might be finite, might be uncountable) of sets where each set is an open set of A and the union of the open sets is equal to A. Can I conclude that A is open?

I am not sure because the sets are open in A, not necessarily open in R. I don't know much about A other than it is some subset of R.

Any help?
 
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You can't conclude that A is open. For a trivial example, take a non-open subset A of R. Note that A is open in itself.
 
If all that you know is that the sets are open in A this can't tell you anything about whether A is open in R. E.g. A is always open in A.
 

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