Sierpiński space

The two-point topological space with open sets ∅, {1}, and {0,1}. It is the smallest nontrivial space that is not discrete and distinguishes its points by open sets.

Topological space: A set equipped with a collection of subsets designated as open, satisfying the topology axioms. Sierpiński space is a finite, minimal example of a topological space.
Specialization order: The partial order obtained from the specialization preorder when that preorder is antisymmetric. The Sierpiński topology directly represents a two-element order through point specialization.
Subobject classifier: An object in category theory that represents subobjects through morphisms into it. Sierpiński space is the topological counterpart of a truth-value object classifying open subspaces.
Kolmogorov space: A topological space in which any two distinct points are distinguished by some open set. Sierpiński space satisfies this separation condition despite lacking stronger separation properties.
Open set: A member of a topology, representing a subset regarded as open under that topology. Only one singleton is open in Sierpiński space, producing its asymmetry.
Continuous function: A function between topological spaces whose inverse images of open sets are open. Maps into Sierpiński space correspond to open subsets of their domain.
Heyting algebra: A bounded lattice with an implication operation characterized by an adjunction to meet. Open subsets of a space form a Heyting algebra, and the two-point case is Sierpiński's opens.
T1 space: A topological space in which every singleton is closed. One singleton in Sierpiński space is not closed, so the space is not T1.
Discrete space: A topological space in which every subset is open. Unlike a discrete space, Sierpiński space does not make both singletons open.
Characteristic function: A function taking one value on a specified subset and another value outside it. A subset's characteristic map is continuous precisely when the selected subset is open, with suitable value assignment.

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