Results on pretrees, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Pretree — Definition 2.1. A set with a ternary betweenness relation satisfying four axioms that hold for the vertices of a tree.
• Four points in a row — Lemma 2.3. Three implications that place four points of a pretree in a row.
• Intervals along a chain — Lemma 3.1. The interval between the ends of a finite sequence of points lies in the union of the intervals between consecutive points.
• Finite pretrees and finite trees — Theorem 4.1. Every finite pretree is a set of vertices of a finite tree with the betweenness of the tree.
• Cycles of adjacent points — Lemma 4.3. Distinct points of a pretree that are cyclically adjacent are pairwise adjacent.
• Directions — Proposition 5.2. A linear subset of a pretree with two chosen points has exactly one direction in which the first precedes the second.
• Intervals are ordered — Corollary 5.3. A closed interval of a pretree is an arc, ordered from one end to the other, and its ends are determined by it.
• Uniqueness of medians — Proposition 6.2. Three points of a pretree have at most one median.
• Half-open intervals — Lemma 1.2. In a dense pretree, two half-open intervals with the same closed end that are equal have the same open end.
• Complete and dense does not imply median — Proposition 3.2. A tripod with a half-open segment next to the centre removed on one leg is a complete dense pretree in which three points have no median and an interval has a cut that is not made at a point.
• Ends of half-open intervals — Lemma 4.1. In a dense median pretree, if the half-open interval from x to y lies in the closed interval from x to z, then y lies in it.
• Cuts are made at points — Theorem 4.3. In a complete dense median pretree, every partition of a closed interval into two arcs is made at a point of the interval.
• Intervals have no gaps — Corollary 4.4. In a complete dense median pretree every nonempty subset of a closed interval has a least upper bound in the interval.
• Gates — Lemma 1.1. In a tree, the path from a vertex to a point of a given path passes through the gate of the vertex, and the path between two vertices with different gates runs through both gates.
• From weakly to strongly dense — Proposition 2.2. In a median pretree the medians of triples from a weakly dense set form a strongly dense set.
• Cantor's theorem — Theorem 3.1. Any two nonempty countable dense totally ordered sets without ends are order isomorphic.
• Order type of intervals — Theorem 4.1. In a complete separable median pretree every interval with distinct ends is order isomorphic to the real interval [0,1].
• Intervals are arcs — Proposition 3.1. Under the assumptions of the theorem, every interval of the pretree with distinct ends is homeomorphic to the real interval [0,1], by an order isomorphism.
• Small intervals — Proposition 4.2. Every neighbourhood of a point contains the intervals from that point to all points of a smaller neighbourhood.
• Countable base — Proposition 4.4. The sets H(p,q) with p, q in a countable strongly dense set form a subbase of the topology, so the space is metrizable.
• Dendrites from pretrees — Theorem 5.1. A compact space with a countable dense subset carrying a complete dense median pretree structure with open sets H(x,y) is a dendrite, and the pretree intervals are its only arcs.
• Flows towards points — Lemma 2.2. A flow is the flow towards a point a exactly when it flows from every other point to a.
• Adding a point for a flow — Proposition 3.1. For a flow on a pretree, adding one terminal point whose betweenness is given by the flow yields a pretree in which the flow is the trace of the flow towards the new point.
• The pretree of a pretree and its flows — Theorem 4.1. A pretree together with the set of all its flows is a pretree in which every flow is terminal and is the trace of the flow towards itself.
• Cofinal arcs — Lemma 2.3. Directed arcs with a common final segment on which their orders agree have the same flow.
• The flow of a directed arc — Proposition 2.4. The relation defined by a directed arc is a flow; on the arc it is the order, and from a point of the arc it reaches exactly the later points of the arc.
• Arcs with the same flow — Lemma 2.6. Two directed arcs with a common point and the same flow are cofinal.
• Restriction to a full subset — Lemma 3.1. If the part of a directed arc inside a full subset is endless, the flow of the arc restricts on the subset to the flow of that part.
• Completeness and suprema — Theorem 4.3. A pretree is complete if and only if every directed arc has a supremum.
• Intervals between flows — Lemma 1.1. Two different flows have a point of the pretree between them; the interval between the flows towards two points is the closed interval between the points.
• Betweenness of flows — Definition 2.1. A flow r lies between flows p and q if r is the flow towards a point between p and q, or r is not a flow towards a point and no point lies both between p and r and between r and q.
• The flows form a pretree — Theorem 3.1. The set of flows of a pretree with this betweenness is a pretree.
• Medians of flows — Theorem 4.1. Every three flows have a median; when no point and none of the three flows serves, it is the flow that goes where two of the three go.
• Completeness — Theorem 5.2. Every arc in the pretree of flows is an interval.
• The completion — Corollary 5.3. Every pretree embeds, with its betweenness, in the complete median pretree of its flows.
• Star — Definition 2.1. A set of at least two points of a pretree that is full and is maximal among the sets with no betweenness relation among their points.
• The flow of a star — Theorem 3.1. From each point the flow of a star runs along the interval to the nearest point of the star; this is a flow, and the star is the set of points from which it runs nowhere.
• Classes and type of a flow — Lemma 4.2. The arcs along which a flow runs fall into classes, each endless or ending in one point; the type of the flow counts the classes of each kind.
• Three kinds of flows — Theorem 5.1. Every flow on a nonempty pretree is exactly one of: the flow towards a point, the flow of a star, the flow of an endless directed arc; the type decides which.
• Points of the pretree are not adjacent — Proposition 1.1. Between the flows towards two different points of a pretree there is always a further flow in the completion.
• Adjacent points of the completion — Theorem 2.3. Two points of the completion are adjacent exactly when one is the flow towards a point a and the other is a different flow that runs nowhere from a; so exactly one of them is a point of the pretree.
• Neighbours and finite intervals — Corollary 2.4. A flow of type (m,n) that is not a flow towards a point has n neighbours; a finite interval of the completion alternates between points of the pretree and other flows.
• Neighbours and branches — Theorem 3.2. The neighbours in the completion of a point a of the pretree correspond one to one to the branches at a, the classes of the relation that a is not between two points.
• Finite intervals are chains — Lemma 1.2. A finite interval of a pretree is a chain of points, each adjacent to the next, in the order of the interval.
• Discrete median pretrees and trees — Theorem 2.1. A pretree with finite intervals in which every three points have a median is the vertex set of exactly one tree; its edges join the adjacent pairs.
• Discrete pretrees in trees — Theorem 3.2. Every pretree with finite intervals is a set of vertices of a tree, with the betweenness of the tree, in which each further vertex is joined to at least three points of the pretree.
• Completion of a finite pretree — Proposition 4.1. The completion of a finite pretree is the vertex set of a finite tree in which every edge joins a point of the pretree to a flow that is not a flow towards a point.
• The quotient pretree — Theorem 2.2. For an equivalence relation with full classes on a pretree, the set of classes is a pretree: a class lies between two others if one of its points lies between them.
• Intervals map onto intervals — Proposition 3.1. The quotient map sends the interval between two points onto the interval between their classes.
• A quotient of a quotient — Proposition 3.3. A full relation on a quotient pulls back to a full relation on the pretree, with the same quotient.
• Quotients of median pretrees — Theorem 4.1. A quotient of a median pretree by a full relation is median, and the quotient map preserves medians.
• Quotients of complete pretrees — Theorem 4.2. A quotient of a complete pretree by a full relation is complete.
• The least codense relation — Proposition 2.3. Among the full relations on a pretree whose quotient is dense there is a least one; its quotient is the dense quotient.
• Rational subsets — Theorem 3.3. Two points are identified in the dense quotient exactly when the interval between them contains no countable linear subset that is densely ordered without ends.
• The derived relation — Lemma 4.2. Identifying points whose classes are joined by a finite interval gives a coarser full relation, equal to the given one exactly when its quotient is dense.
• Reaching the least codense relation — Theorem 5.2. Starting from equality and repeating the derived relation along the ordinals, with unions at limits, one reaches the least codense relation.
• Rational subsets in the completion — Proposition 6.2. Two points of the completion are identified in its dense quotient exactly when no rational subset consisting of points of the pretree lies between them.
• The dense quotient of the completion — Corollary 6.3. The dense quotient of the completion of a pretree is a complete dense median pretree, and it contains the dense quotient of the pretree.
• Two cut points — Lemma 1.2. Two different cut points of a connected Hausdorff space split it into three nonempty open sets and the two points, arranged in a row.
• The pretree of a connected space — Theorem 2.1. A connected Hausdorff space is a pretree when a point lies between two others if it has a separation with one of them on each side; the points between others are the cut points.
• Branches meet connected sets in connected sets — Lemma 3.2. A closed set with at least two points and one boundary point meets every closed connected set in a connected set.
• Intersections of compact connected sets — Lemma 4.2. In a Hausdorff space, the intersection of a family of compact connected sets that is closed under finite intersections is connected.
• Intersections of branches — Proposition 5.2. Every intersection of branches of a connected Hausdorff space meets each compact connected set in a connected set; in a compact space it is connected.
• The map to the completion and betweenness — Lemma 1.2. For a pretree M with a subset T, the map that sends a point of M to the flow of T towards it sends points of T between two points to points between the images, and conversely.
• Images of the points on the side of a flow — Proposition 2.3. A point of M that lies, for every point a of T, on the side of a on which a flow p lies is mapped to p or to a neighbour of p in the completion.
• The completion of a set of cut points — Theorem 3.2. For a compact connected Hausdorff space and a set T of cut points, every point of the completion of T is the image of a point of the space or adjacent to the image of a point of T; the set of points on the side of a flow is nonempty, closed and connected.
• Fibres of the quotient map — Proposition 1.2. Two points of a compact connected Hausdorff space have the same image in the quotient exactly when no rational subset of the chosen cut points lies between them.
• Preimages of preclosed sets — Proposition 2.4. The preimage in the space of a preclosed subset of the completion of the set of cut points is closed and connected.
• The quotient is a continuum — Theorem 3.4. The quotient is a compact connected Hausdorff space, and the preimage of every closed connected subset is closed and connected.
• The dendrite quotient — Theorem 4.1. If the space has a countable dense subset, the quotient is a dendrite; its pretree intervals are its arcs, and its pretree is the pretree of its topology.
• Spaces in which points are separated by points — Corollary 4.2. A compact connected Hausdorff space with a countable dense subset in which any two points are separated by a third point is a dendrite.

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