Projection from Product Topology is Open

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Theorem

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $T = \left({T_1 \times T_2, \tau}\right)$ be the topological product of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$..

Let $\operatorname{pr}_1: T \to T_1$ and $\operatorname{pr}_2: T \to T_2$ be the first and second projections from $T$ onto its factors.


Then both $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are open.


General Result

Let $\left \langle T_i \right \rangle_{i \mathop \in I} = \left \langle \left({S_i, \tau_i}\right) \right \rangle_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space.

Let $\tau$ denote the Tychonoff topology on $S$.

Let $\operatorname{pr}_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$.


Then $\operatorname{pr}_i$ is open for all $i \in I$.


Proof

First, we prove that $\operatorname{pr}_1$ is open.


If $U \in \tau$, it follows from the definition of Tychonoff topology that $U$ can be expressed as:

$\displaystyle U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \operatorname{pr}_{i_{k,j} }^{-1} \left({ U_{k,j} }\right)$

where $J$ is an arbitrary index set, $n_j \in \N$, $i_{k,j} \in \left\{ {1,2}\right\}$, and $U_{k,j} \in \tau_{i_{k,j} }$.

For all $i \in \left\{ {1, 2}\right\}$, define $V_{i, k, j} \in \tau_{i}$ by $V_{i, k, j} = U_{k, j}$ if $i = i_{k, j}$, and $V_{i, k, j} = S_i$ if $i \ne i_{k, j}$.

By definition of projection:

$\displaystyle \operatorname{pr}_{i_{k, j} }^{-1} \left({ U_{k, j} }\right) = \prod_{i \mathop = 1}^2 V_{i, k, j}$

Then:

\(\displaystyle \operatorname{pr}_1 \left({ U }\right)\) \(=\) \(\displaystyle \bigcup_{j \mathop \in J} \operatorname{pr}_1 \left({ \bigcap_{k \mathop = 1}^{n_j} \operatorname{pr}_{i_{k,j} }^{-1} \left({ U_{k,j} }\right) }\right)\) Image of Union under Relation: Family of Sets
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{j \mathop \in J} \operatorname{pr}_1 \left({ \bigcap_{k \mathop = 1}^{n_j} \prod_{i \mathop = 1}^2 V_{i, k, j} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{j \mathop \in J} \operatorname{pr}_1 \left({ \prod_{i \mathop = 1}^2 \bigcap_{k \mathop = 1}^{n_j} V_{i, k, j} }\right)\) Cartesian Product of Intersections
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} V_{1,k,j}\) Definition of Projection

As $\displaystyle \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} V_{1,k,j} \in \tau_1$, it follows that $\operatorname{pr}_1$ is open.

The proof for $\operatorname{pr}_2$ is symmetrical.

$\blacksquare$