Baire Space (set Theory) - Properties

Properties

The Baire space has the following properties:

  1. It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of the term.
  2. It is zero dimensional and totally disconnected.
  3. It is not locally compact.
  4. It is universal for Polish spaces in the sense that it can be mapped continuously onto any non-empty Polish space. Moreover, any Polish space has a dense Gδ subspace homeomorphic to a Gδ subspace of the Baire space.
  5. The Baire space is homeomorphic to the product of any finite or countable number of copies of itself.

Read more about this topic:  Baire Space (set Theory)

Famous quotes containing the word properties:

    “A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
    —Ralph Waldo Emerson (1803–1882)

    “The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
    —John Locke (1632–1704)