# 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 $z:A\to {\cup}_{\alpha}{Z}_{\alpha}$ 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

$${\pi}_{X,\alpha}:X\to {X}_{\alpha},{\pi}_{Z,\alpha}:Z\to {Z}_{\alpha}$$ |

be the canonical projections defined for $X$ and $Z$. The subbases (http://planetmath.org/Subbasis) for $X$ and $Z$ are given by

${\beta}_{X}$ | $=$ | $\mathrm{\{}{\pi}_{X,\alpha}^{-1}(U):\alpha \in A,U\in \tau ({X}_{\alpha})\},$ | ||

${\beta}_{Z}$ | $=$ | $\mathrm{\{}{\pi}_{Z,\alpha}^{-1}(U):\alpha \in A,U\in \tau ({Z}_{\alpha})\},$ |

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

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

∎

Title | product topology and subspace topology |
---|---|

Canonical name | ProductTopologyAndSubspaceTopology |

Date of creation | 2013-03-22 15:35:33 |

Last modified on | 2013-03-22 15:35:33 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 6 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 54B10 |