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.