Section 19: Problem 4 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

Show that $(X_{1}\times\cdots\times X_{n-1})\times X_{n}$ is homeomorphic with $X_{1}\times\cdots\times X_{n}$ .

The homeomorphism is $f(((x_{1},...,x_{n-1}),x_{n}))=(x_{1},...,x_{n})$ . Indeed, $U$ is open in $(X_{1}\times\cdots\times X_{n-1})\times X_{n}$ iff every point $(x_{1}\times\cdots\times x_{n-1})\times x_{n}$ of $U$ has a basis neighborhood $V\times U_{n}\subset U$ where $V$ is open in $X_{1}\times\cdots\times X_{n-1}$ iff every point $(x_{1}\times\cdots\times x_{n-1})\times x_{n}$ of $U$ has a basis neighborhood $(U_{1}\times\cdots\times U_{n-1})\times U_{n}\subset U$ iff every point $x_{1}\times\cdots\times x_{n}$ of $f(U)$ has a basis neighborhood $U_{1}\times\cdots\times U_{n}\subset f(U)$ .

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Section 19: Problem 4 Solution

Section 19: Problem 4 Solution