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Defn: If $X$ is an ordered set and $a \in X$, then the following are rays in $X$:  

\centerline{$(a, +\infty) = \{x ~|~ x > a\}, ~(-\infty, a) = \{x ~|~ x < a\}$,}

\centerline{$[a, +\infty) = \{x ~|~ x \geq a\}, ~(-\infty, a] = \{x ~|~ x \leq a\}.$}

Lemma:  The collection of all open rays is a subbasis for the order topology.

\vskip 6pt
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15:  The Product Topology

Let ${\cal T}_X$ denote the topology on $X$ and ${\cal T}_Y$ denote the topology on $Y$.

Defn:  Let $X$ and $Y$ be topological Spaces.  The {\bf product topology} on $X \times Y$ is the topology having as basis ${\cal B } = \{U \times V ~|~ U \in {\cal T}_X , V \in {\cal T}_Y \}$.

Thm 15.1:  If ${\cal B}_X$ is a basis for the topology of $X$ and  ${\cal B}_Y$ is a basis for the topology of $Y$,  then ${\cal D} = \{ U \times V ~|~ U \in {\cal B}_X , V \in {\cal B}_Y \}$ is a basis for the topology of  $X \times Y$.

Ex. 1:  If $R$ has the standard topology, the product topology on $R \times R$ is the standard topology on $R^2$.

Defn:  Let $\pi_1: X_1 \times X_2 \rightarrow X_1$, $\pi_1(x_1, x_2) = x_1$.
\break
 $\pi_1$ is the projection of $X_1 \times X_2$ onto the first
component.

\f

Note:  If $U \subset X_1$, then $\pi_{1}^{-1}(U) = U \times X_2$.  Thus
if
$U$ is open in $X_1$, then $\pi_{1}^{-1}(U)$ is open in $ X_1 \times
X_2$

\f

Note:  $\pi_{1}^{-1}(U) \cap \pi_{2}^{-1}(V) = U \times V$

\f

Thm 15.2:  The collection \hskip -20pt $${\cal S} = \{\pi_{1}^{-1}(U)
~|~ U
\hbox{ open in}
X
\} \cup \{\pi_{2}^{-1}(V) ~|~ V \hbox{ open in} Y \}$$ is a subbasis
for the
product topology on $X \times Y$.


\f\f
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HW p. 91:  4, 6

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