Consider the field C together with the Euclidean topology. Let K be

a proper subfield of C that is not contained in R.

Which of the following statements is necessarily true ?

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Correct option is A

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$$\text{(iii) As } K \text{ is a proper subfield, we can find an element } K_1 \in \mathbb{C} \text{ but } K_1 \notin K. \\\text{So, the 3rd option is incorrect.} \\\text{(iv) As } \overline{K} = \mathbb{C} \; (\text{since } K \text{ is dense in } \mathbb{C}), \\\text{the smallest closed subset of } K \text{ will be a field.} \\\text{(Option 4 is incorrect.)}$$

$$\text{(iii) As } K \text{ is a proper subfield, we can find an element } K_1 \in \mathbb{C} \text{ but } K_1 \notin K. \\\text{So, the 3rd option is incorrect.} \\\text{(iv) As } \overline{K} = \mathbb{C} \; (\text{since } K \text{ is dense in } \mathbb{C}), \\\text{the smallest closed subset of } K \text{ will be a field.} \\\text{(Option 4 is incorrect.)}$$

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