Section 29: Problem 6 Solution
Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
The circle without a point is homeomorphic to the real line. Now using two facts, first, that the one-point compactification is unique up to a homeomorphism, and, second, that the compactification of the punctured circle is the whole circle (in fact, showing this is quite similar to the next exercise), we get the result.