Chapter 7, Product and Quotient Spaces Video Solutions, Principles of topology

Problem 1

Let $O_i$ and $U_i$ be subsets of a set $X_i$ for $i=1,2$. Prove that
$$
\left(O_1 \times O_2\right) \cap\left(U_1 \times U_2\right)=\left(O_1 \cap U_1\right) \times\left(O_2 \cap U_2\right) .
$$
Generalize this result to the case of $n$ sets $X_1, X_2, \ldots, X_n$.

Vysakh M

Numerade Educator

Problem 2

Show that the set $B$ in the definition of product topology is actually a basis, as claimed in the definition.

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Problem 3

Let $X_1, X_2$, and $X_3$ be spaces.
(a) Prove that $\left(X_1 \times X_2\right) \times X_3$ is homeomorphic to $X_1 \times\left(X_2 \times X_3\right)$.
(b) Prove that $X_1 \times X_2$ is homeomorphic to $X_2 \times X_1$.

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01:45

Problem 4

Prove Theorems 7.4,7.5, and 7.6.

J Hardin

Numerade Educator

Problem 5

Use the Alexander Subbasis Theorem to give a different proof of Theorem 7.7.

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Problem 6

Knowing that a subset $A$ of $\mathrm{R}$ is compact if and only if it is closed and bounded, use product space ideas to prove the same characterization of the compact subsets of $\mathrm{R}^n$.

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Problem 7

Let $X_1$ and $X_2$ be spaces with subsets $A \subset X_1$ and $B \subset X_2$. In the product space $X_1 \times$ $X_2$, prove that
(a) $\overline{A \times B}=\bar{A} \times \bar{B}$.
(b) int $(A \times B)=\operatorname{int} A \times$ int $B$.

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Problem 8

Let $X$ denote the real line with the half-open interval topology of Example 4.3 .4 and let $Y=X \times X$.
(a) Prove that $\mathscr{B}=\{[a, b) \times[c, d): a, b, c, d \in \mathrm{R}, a<c, b<d\}$ is a basis for $Y$.
(b) Show that $Y$ is separable.
(c) Show that the line $y=-x+1$ is a non-separable subspace of $Y$.
(d) Show that $X$ is Lindelöf but $Y$ is not. (Hint: The Lindelöf property is inherited by closed subspaces. Show that the line of part (c) is closed and not Lindeloff.)

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Problem 9

Prove that the product of a finite family of locally compact spaces is locally compact.

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01:56

Problem 10

This exercise is intended to show that a result like Theorem 7.1 does not hold for functions $g: \Pi X_i \rightarrow Y$ when the domain is a product space instead of the range.
Definition: Let $X_1$ and $X_2$ be spaces and $g: X_1 \times X_2 \rightarrow Z$ a function from $X_1 \times X_2$ to a space $Z$. Then $g$ is continuous in the first variable provided that for each $y$ in $X_2$, the function $g(\cdot, y): X_1 \rightarrow Z$, whose value at $x$ is $g(x, y)$, is continuous. Continuity in the second variable is defined in the analogous way.
Show that the function $f: \mathbb{R} \times \mathrm{R} \rightarrow \mathrm{R}$ defined by
$$
f(x, y)= \begin{cases}x y /\left(x^2+y^2\right) \cdot & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}
$$
is continuous in the first variable and continuous in the second variable but not continuous at $(0,0)$.

James Kiss

Numerade Educator

Problem 11

Definition: In a product space $X \times X$, the set $\{(x, x): x \in X\}$ is called the diagonal. Prove that a space $X$ is Hausdorff if and only if the diagonal of $X \times X$ is a closed set.

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