Continuous Surjection~ nonexistence (TIFR 2013 problem 26)

Question:

True/False?

There exists a continuous surjective map from the complex plane onto the non-zero reals.

Hint:

Search for topological invariants.

Discussion:

Under a continuous function, connected set must go to connected set. The complex plane \(\mathbb{C}\) is connected.

It’s image must be connected.

\(\mathbb{R}-0\) is not connected.

So the statement is False.

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