[:finite product topology:] [:product topology:] • A product topology on $X$ and $Y$ (both with an implciit topology) is the topology $X \times Y$ generated by the topological basis $\mathscr B$ given by sets of the form $U \times V$ where $U$ is open in $X$ and $V$ is open in $Y$ • Intuition: consider $X = Y = \mathbb R_\text{std}$. Elements of $\mathscr B$ are then open rectangles with closed horizontal and vertical stripes taken out (wish I could draw on Roam lol) • Note: This definition generalizes to to finite products of topologies, but not to infinite products: then there are multiple canonical definitions (infinite product topology, box topology) • Note that $\mathscr B$ is not itself a topology. Let $X = Y = \mathbb R_\text{std}$. Then $(0\dots 2)^2$ and $(1\dots 3)^2$ are both in $\mathscr B$, and so is their intersection, but their union is not • Here I use $(a \dots b)$ to mean the interval $(a, b)$, to avoid ambiguous notation • Proof that $\mathscr B$ is a basis: • Condition 1: $\mathscr B$ covers $X \times Y$ • Intuition: true since $\mathscr T_X$ covers $X$ and $\mathscr T_Y$ covers $Y$ • Take an $\langle x, y\rangle \in X \times Y$. WTS that $\langle x, y\rangle$ is in some basis element • Note that there is some $U_x$ open in $X$ containing $x$, and some $U_y$ open in $Y$ containing $y$ • Note that $U_x \times U_y$ contains $\langle x, y\rangle$ and is a basis element. Done. • Condition 2: given $B_1, B_2 \in \mathscr B$ and $x \in B_1 \cap B_2$, exists some $B' \in \mathscr B$ s.t. $B' \subseteq B_1 \cap B_2$ and $x \in B'$ • Take $B_1, B_2 \in \mathscr B$ and $x \in B_1 \cap B_2$ • Let $B' = B_1 \cap B_2$ • Then clearly $x \in B'$ • What remains to show is that $B'$ is a basis element. • Write basis elements $B_1$ as $U_1 \times V_1$ and $B_2$ as $U_2 \times V_2$ • Then $B' = (U_1 \times V_1) \cap (U_2 \times V_2) = (U_1 \cap U_2) \times (V_1 \cap V_2)$$U_1 \cap U_2$ and $V_1 \cap V_2$ are both open in their respective spaces, so $B'$ is a basis element • • Note that if $\mathscr B$ is a basis for $\mathscr T_X$ and $\mathscr C$ is a basis for $\mathscr T_Y$ then the collection $\mathscr D = \{ B \times C : B \in \mathscr B, C \in \mathscr C \}$ is a basis for $X \times Y$ • Proof: elided (for now?) • • The functions $\pi_1 = (x, y) \mapsto x$ and $\pi_2 = (x, y) \mapsto y$ are called projections of $X \times Y$ • Then $\pi_1^{-1}(x_0) = \{ \langle x, y \rangle \in X \times Y : x = x_0 \}$ • Read: $\pi_1^{-1}(x)$ are all those 2-tuples where the first element is $x$ • Intuition: for $X = Y = \mathbb R$, $\pi_1^{-1}(1)$ is the vertical line $x = 1$ • And for the interval $I = (x_0, x_f)$, $\pi_1^{-1}(I)$ is the vertical strip from $x = x_0$ through $x = x_f$ • Read: $\pi_1^{-1}(U)$ are all those 2-tuples where the first elements is in $U$ • ![](https://firebasestorage.googleapis.com/v0/b/firescript-577a2.appspot.com/o/imgs%2Fapp%2Fplace%2Fiu6t-mSzvo.png?alt=media&token=48969a03-22ca-43ba-af8c-3e7fbe388600) • We can define a topological subbasis $\mathcal S$ for $X \times Y$ as follows: • Let $\text{Vert} = \{ \pi_1^{-1}(V) : V \text{ open in } Y \}$ • Let $\text{Horiz} = \{ \pi_2^{-1}(U) : U \text{ open in } X \}$$\mathcal S = \text{Horiz} \cup \text{Vert}$ • I will show that $\mathcal S$ is a subbasis for $X \times Y$ by showing that elements of the previously-discussed $\mathcal B$ are intersections of $\mathcal S$. Also see subbasis to basis. • Take some $B \in \mathcal B$ • Then $B = U \times V$ for some $U$ open in $X$ and $V$ open in $Y$ • Claim: $B = \pi_1^{-1}(U) \cap \pi_2^{-1}(V)$ • Will prove by showing $\langle x, y \rangle \in B \iff \langle x, y \rangle \in \pi_1^{-1}(U) \cap \pi_2^{-1}(V)$$(\Longrightarrow)$ • Take $\langle x, y \rangle \in B$ • Then $x \in U$ and $y \in V$ • Since $x \in U$ then $\langle x, y \rangle \in \pi_1^{-1}(U)$ • Since $y \in V$ then $\langle x, y \rangle \in \pi_2^{-1}(V)$ • Thus $\langle x, y \rangle \in \pi_1^{-1}(U) \cap \pi_2^{-1}(V)$$(\Longleftarrow)$ • Take $\langle x, y \rangle \in \pi_1^{-1}(U) \cap \pi_2^{-1}(V)$ • Then $x \in U$ and $y \in V$ • So $\langle x, y \rangle \in U \times V$ Referenced by: