Continuity in Topology
Preimages of open sets are open.
Key idea
Preimages of open sets are open.
In depth
f is continuous if the preimage of every open set is open. It needs no distances and agrees with ε–δ continuity for metric spaces. Compositions of continuous functions are continuous.
Worked example
f(x) = x² : f⁻¹((1, 4)) = (−2, −1) ∪ (1, 2), open.
Real-world uses
- Describing continuous maps between shapes, such as deforming one object into another
Common mistake
Using images of open sets rather than preimages.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Continuity in Topology?
Preimages of open sets are open. It is a topic at the expert level in Analysis & Topology.
Where is Continuity in Topology used in real life?
Real-world examples: Describing continuous maps between shapes, such as deforming one object into another.
What should I learn before Continuity in Topology?
Make sure you are comfortable with Bases and Subbases, the previous topic in Analysis & Topology.
What is a common mistake with Continuity in Topology?
Using images of open sets rather than preimages.
How can I practise Continuity in Topology?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Continuity in Topology?
Move on to Homeomorphisms next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.