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I want a counter example that why is interior of union of arbitrary family of sets not a subset of union of arbitrary family of interior sets

2 Answers 2

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$\mathbb{Q}$ and $\mathbb{Q}^c$.

or

$(1,2]$ and $[2,3]$.

For infinite case let the rests be nulls.

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    But here are only two sets. – 2017-01-22
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    How many do you want, you said **arbitrary family** – 2017-01-22
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    What if arbitrarily many intervals are to be used? – 2017-01-22
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    @VikasSharma Added, check it – 2017-01-22
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In ${\mathbb{R}}$

Let $S_x = {x}, x \in \mathbb{R}$

Interior of any S is empty set

Interior of union is ${\mathbb{R}}$