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:

    “There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
    —Adrienne Rich (b. 1929)

    “Every word was once a poem. Every new relation is a new word.”
    —Ralph Waldo Emerson (1803–1882)

    “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)