Topological Spaces: Continuity Without Distance
Declare which sets count as open and continuity can be defined with no distance anywhere in sight. One set can carry several topologies, and the choice decides which functions are continuous.
What a learner can do afterwards
- Check the axioms for a proposed collection of open sets
- Define continuity by preimages and reconcile it with the epsilon-delta version
- Give two topologies on one set and a function continuous for one but not the other
1 · Read
A topology declares which sets count as open. It must hold the empty set and the whole set, stay closed under arbitrary unions, and stay closed under finite intersections. On the set with points 1 and 2, the collection holding the empty set, the set with 1 alone, and the whole set passes all three tests.
Continuity then needs no ruler: a function is continuous when the preimage of every open set is open. Images go the wrong way and need not cooperate, so never test continuity with images of opens. Unions are unrestricted while intersections stay finite, which is the asymmetry to memorize.
On the real line this new wording agrees with epsilon-delta continuity. Every open set there is built from open intervals, so controlling outputs by restricting inputs becomes a preimage condition. Learn both dialects so you recognize the same idea far from any metric.
One set can carry several topologies, and the choice decides which functions are continuous. The identity map from a set with the discrete topology to the same set with the indiscrete topology is continuous, while the reverse map is not. Restricting a domain works the same way: deleting bad points can create continuity.
A topology is three axioms on open sets, continuity means open preimages, and the same function can change status when the topology changes.
More to read
2.4 Continuity
Continuity means small input changes give small output changes, with no jumps or tears. The open-set version says the same thing without rulers: preimages of opens are open. On the real line the two agree, since every open set is built from intervals. Learning both dialects lets you recognize the same idea far from any metric.
2.2 The Limit of a Function
Limits describe approach: values near a point force function values near a target. Continuity adds that the target equals the actual value there. The limit laws then compute with continuous functions almost automatically. This progression from approach to contact is the seed of topological thinking.
3.2 Domain and Range
A function pairs each input with exactly one output, and domain and range name the two sides. Restricting the domain can create continuity where none existed, by deleting the bad points. Comparing topologies on one set is the same game with openness instead of definedness. Notation that tracks domains carefully is topology in disguise.
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Where it sits
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Optional connections
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.