Quotient Space - Compatibility With Other Topological Notions

Compatibility With Other Topological Notions

  • Separation
    • In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of X need not be inherited by X/~, and X/~ may have separation properties not shared by X.
    • X/~ is a T1 space if and only if every equivalence class of ~ is closed in X.
    • If the quotient map is open, then X/~ is a Hausdorff space if and only if ~ is a closed subset of the product space X×X.
  • Connectedness
    • If a space is connected or path connected, then so are all its quotient spaces.
    • A quotient space of a simply connected or contractible space need not share those properties.
  • Compactness
    • If a space is compact, then so are all its quotient spaces.
    • A quotient space of a locally compact space need not be locally compact.
  • Dimension
    • The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.

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