General topology
Properties

If f: X → Y and g: Y → Z are continuous, then so is the composition g ∘ f: X → Z. If f: X → Y is continuous and

The possible topologies on a fixed set X are partially ordered: a topology τ is said to be coarser than another topology τ (notation: τ ⊆ τ) if every open subset with respect to τ is also open with respect to τ. Then, the identity map

is continuous if and only if τ ⊆ τ (see also comparison of topologies). More generally, a continuous function

stays continuous if the topology τ is replaced by a coarser topology and/or τ is replaced by a finer topology.
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