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MathidiaMathematics Academy
Topology
Lesson 12 of 75
Easy practice

Topological continuity

Objective. Use inverse images to prove continuity.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A function f:X→Y between topological spaces is topologically continuous if f⁻¹(U) is open in X for every open set U in Y, generalising the ε-δ definition without needing a metric. This preimage criterion makes continuity easy to verify using a basis: it suffices to check preimages of basis elements are open. Composition of continuous functions is continuous, and homeomorphisms are continuous bijections with continuous inverse, meaning they preserve all topological properties. For instance, the identity map from the discrete topology to any topology on the same set is always continuous since every preimage is automatically open in the discrete space.

Worked example

Determine whether f:ℝ→ℝ (usual topology) given by f(x)=⌊x⌋ (floor) is topologically continuous.

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  1. 1Take open set U=(0.5,1.5) in the codomain
  2. 2f⁻¹(U) is the set of x with ⌊x⌋ in (0.5,1.5), i.e. ⌊x⌋=1, so x∈[1,2)
  3. 3[1,2) is not open in ℝ since it includes its left endpoint without an open neighbourhood
  4. 4Since a preimage of an open set fails to be open, f is not continuous

Answer: No, f is not continuous

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Common mistake. Students check continuity by looking at images of open sets rather than preimages; f can map open sets to non-open sets and still be continuous, as with f(x)=x² on ℝ sending the open interval (−1,1) to [0,1), which is not open, yet f is still continuous.

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