For each [tex]n \in \omega[/tex], let [tex]X_n[/tex] be the set [tex]\{0, 1\}[/tex], and let [tex]\tau_n[/tex] be the discrete topology on [tex]X_n[/tex]. For each of the following subsets of [tex]\prod_{n \in \omega} X_n[/tex], say whether it is open or closed (or neither or both) in the product topology.(adsbygoogle = window.adsbygoogle || []).push({});

(a) [tex]\{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}[/tex]

(b) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n) = 0 \}[/tex]

(c) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}[/tex]

(d) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}| = 5 \}[/tex]

(e)[tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}|\leq5 \}[/tex]

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Discrete topology, product topology

Loading...

Similar Threads for Discrete topology product | Date |
---|---|

I Metrics and topologies | Friday at 12:28 PM |

Computing a discrete surface integral of a scalar function | Oct 24, 2013 |

Plausibility of a Discrete Point Manifold | Jul 3, 2013 |

Will limit of discrete steps give Pythagoras theorem? | Apr 3, 2011 |

Discrete quotient group from closed subgroup | Mar 28, 2011 |

**Physics Forums - The Fusion of Science and Community**