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.
Read more about this topic: Quotient Space
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