Part CCSIR NET December 2023projection-is-open-map

Projection is open map

Let $p : R^{2} \to R$ be the function p(x, y) = x. Which of the following statements are true?

A.Let $A_{1} =$ { $x^{2} + y^{2} < 1$ }. For each $γ \in p (A_{1})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{1})$ .
B.Let $A_{2} =$ { $x^{2} + y^{2} \leq 1$ }. For each $γ \in p (A_{2})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{2})$ .
C.Let $A_{3} =$ {xy = 0}. For each $γ \in p (A_{3})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{3})$ .
D.Let $A_{4} =$ {xy = 1}. For each $γ \in p (A_{4})$ there exists $ε > 0$ with $(γ - ε, γ + ε) \subseteq p (A_{4})$ .

The answer and working come with the PYQ pack.

The trap it tests

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

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