Which of the following statements is true?

Question

Question

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)

There are at most countably many continuous maps from \(\mathbb{R}^2\) to \(\mathbb{R}\).
There are at most finitely many continuous surjective maps from \(\mathbb{R}^2\) to \(\mathbb{R}\).
There are infinitely many continuous injective maps from \(\mathbb{R}^2\) to \(\mathbb{R}\).
There are no continuous bijective maps from \(\mathbb{R}^2\) to \(\mathbb{R}\).

Answer 1

Explanation:

Let f: ℝ² → ℝ is defined by

f(x, y) = cx, c ∈ ℝ\{0} then f is continuous function.

(1) and (2) are false.

If possible let there are infinitely many continuous injective maps f: ℝ2 → ℝ.

Then it will map a connected set to a connected set.

If we consider f such that f(0) = c, c ∈ ℝ\{0} then

f(ℝ\{0}) = (-∞, c) ∪ (c, ∞), which is not connected. So we are getting a contradiction.

(3) is false.

Hence option (4) is correct

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