Product Topology - Relation To Other Topological Notions

Relation To Other Topological Notions

  • Separation
    • Every product of T0 spaces is T0
    • Every product of T1 spaces is T1
    • Every product of Hausdorff spaces is Hausdorff
    • Every product of regular spaces is regular
    • Every product of Tychonoff spaces is Tychonoff
    • A product of normal spaces need not be normal
  • Compactness
    • Every product of compact spaces is compact (Tychonoff's theorem)
    • A product of locally compact spaces need not be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).
  • Connectedness
    • Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
    • Every product of hereditarily disconnected spaces is hereditarily disconnected.

Read more about this topic:  Product Topology

Famous quotes containing the words relation to, relation and/or notions:

    “Hesitation increases in relation to risk in equal proportion to age.”
    —Ernest Hemingway (1899–1961)

    “There is a certain standard of grace and beauty which consists in a certain relation between our nature, such as it is, weak or strong, and the thing which pleases us. Whatever is formed according to this standard pleases us, be it house, song, discourse, verse, prose, woman, birds, rivers, trees, room, dress, and so on. Whatever is not made according to this standard displeases those who have good taste.”
    —Blaise Pascal (1623–1662)

    “The mass believes that it has the right to impose and to give force of law to notions born in the café.”
    —José Ortega Y Gasset (1883–1955)