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# product topology and subspace topology

Let $X_{\alpha}$ with $\alpha\in A$ be a collection of topological spaces, and let $Z_{\alpha}\subseteq X_{\alpha}$ be subsets. Let

$X=\prod_{{\alpha}}X_{\alpha}$ |

and

$Z=\prod_{{\alpha}}Z_{\alpha}.$ |

In other words, $z\in Z$ means that $z$ is a function $z\colon A\to\cup_{\alpha}Z_{\alpha}$ such that $z(\alpha)\in Z_{\alpha}$ for each $\alpha$. Thus, $z\in X$ and we have

$Z\subseteq X$ |

as sets.

###### Theorem 1.

The product topology of $Z$ coincides with the subspace topology induced by $X$.

###### Proof.

Let us denote by $\tau_{X}$ and $\tau_{Z}$ the product topologies for $X$ and $Z$, respectively. Also, let

$\pi_{{X,\alpha}}\colon X\to X_{\alpha},\quad\pi_{{Z,\alpha}}\colon Z\to Z_{\alpha}$ |

be the canonical projections defined for $X$ and $Z$. The subbases for $X$ and $Z$ are given by

$\displaystyle\beta_{X}$ | $\displaystyle=$ | $\displaystyle\{\pi_{{X,\alpha}}^{{-1}}(U):\alpha\in A,U\in\tau(X_{\alpha})\},$ | ||

$\displaystyle\beta_{Z}$ | $\displaystyle=$ | $\displaystyle\{\pi_{{Z,\alpha}}^{{-1}}(U):\alpha\in A,U\in\tau(Z_{\alpha})\},$ |

where $\tau(X_{\alpha})$ is the topology of $X_{\alpha}$ and $\tau(Z_{\alpha})$ is the subspace topology of $Z_{\alpha}\subseteq X_{\alpha}$. The claim follows as

$\beta_{Z}=\{B\cap Z:B\in\beta_{X}\}.$ |

∎

## Mathematics Subject Classification

54B10*no label found*

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