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.

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