Section 18: Problem 3 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

Let $X$ and $X'$ denote a single set in the two topologies $\mathcal{T}$ and $\mathcal{T}'$ , respectively. Let $i:X'\rightarrow X$ be the identity function.

(a) Show that $i$ is continuous $\Leftrightarrow$ $\mathcal{T}'$ is finer than $\mathcal{T}$ .

(b) Show that $i$ is a homeomorphism $\Leftrightarrow$ $\mathcal{T}'=\mathcal{T}$ .

(a) Both mean that "every set open in $\mathcal{T}$ is open in $\mathcal{T}'$ ".

(b) Follows from (a), as $i^{-1}$ is the same identity function.

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Section 18: Problem 3 Solution

Section 18: Problem 3 Solution