What makes R^2 different from R? Unless I'm mistaken (which is unfortunately quite possible), R^2 has the "same number" of points as R.(adsbygoogle = window.adsbygoogle || []).push({});

I'm thinking that what makes R^2 different from R is the topology, which is why I picked this forum. But what sort of conditions do we really need to make R^2 distinguishable from R? Having more of a physics background than a math background, I think of imposing a metric on the space, though I'm not sure if this is overkill. Is there a less stringent notion that we can use to distinguish R^2 from R?

I'm also a bit confused over what's needed to impose a metric. Under metric topologies, the only requirement listed on mathworld,

http://mathworld.wolfram.com/MetricTopology.html

is that the space be Hasdorff. But there are also some entries about T1 spaces and T2 spaces. I'm not clear about what the difference between them is supposed to be, but it mentions that T1 spaces are supposed to be metrizable, and nothing is said one way or the other about the metrizability of T2 spaces

http://mathworld.wolfram.com/T1-Space.html http://mathworld.wolfram.com/T2-Space.html

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

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!

# Dimension in mathematics

Loading...

Similar Threads - Dimension mathematics | Date |
---|---|

B Reverse Mathematics Book | Feb 20, 2018 |

A Markov Chain as a function of dimensions | Aug 10, 2017 |

Dimension of a graphical model | Oct 16, 2014 |

Is it possible to measure Hausdorff dimension for real world objects? | Sep 13, 2014 |

Hausdorff dimension of Hofstadter's Butterfly? | May 1, 2014 |

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