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SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties SAM III General Topology Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Lecture 3 SAM — Seminar in Abstract Mathematics [Version 20130224] is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties 1 Connectedness Discrete and indiscrete spaces Definition Some properties Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness 2 Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Discrete and indiscrete spaces SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties Definition A topological space (X , τ ) is said to be discrete if τ is the finest topology on X , and indiscrete if τ is the coarsest topology on X . Maps from a discrete space to an indiscrete space Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Show that any map from a discrete space to an indiscrete space is continuous. Constructions with discrete and indiscrete spaces Examine stability of the classes of discrete and indiscrete spaces under formation of subspaces, products, sums, and quotient spaces. Definition of connectedness SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties Definition A topological space (X , τ ) is said to be connected if the image of any continuous map (X , τ ) → 2, where 2 is a two-point set with discrete topology, is a singleton. A disconnected space is one which is not connected. Characterization of connected spaces via open subsets Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Show that a topological space (X , τ ) is connected if and only if it is not empty and does not contain two nonempty disjoint open sets A and B such that A ∪ B = X . Characterization of connected spaces via sum of spaces Show that a topological space (X , τ ) is connected if and only if it is not empty and cannot be represented as a sum of two nonempty spaces. Some properties of connected spaces SAM III General Topology Lecture 3 Which discrete/indiscrete spaces are connected? Characterize discrete/indiscrete connected spaces. Contents Connectedness Discrete and indiscrete spaces Definition Some properties Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Reflection of disconnectedness Show that if f : (X , σ) → (Y , τ ) is a continuous surjection and (Y , τ ) is disconnected, then (X , σ) is also disconnected. Deduce from this that if two spaces are homeomorphic then either both of them are connected or both of them are disconnected. Connected preorders Characterize those preorders whose corresponding Alexandroff space is connected. Definition and basic properties SAM III General Topology Lecture 3 Definition A cut of a linearly ordered set (L, 6) is a pair (A, B) of subsets of L such that L = A ∪ B and every element of A is strictly less than every element of B. Contents Connectedness Discrete and indiscrete spaces Definition Some properties Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Cuts via down-closed sets Show that a pair (A, B) is a cut of a linearly ordered set (L, 6) if and only if A and B are disjoint with L = A ∪ B, and A is down-closed, i.e. ∀a∈A ∀x∈L (x 6 a ⇒ x ∈ A). Conclude that there is a one-to-one correspondence between cuts of (L, 6) and down-closed subsets of L. Dualize these observations (use up-closed as the dual of down-closed). Types of cuts Show that for any cut (A, B) of a linearly ordered set (L, 6), the set A has a supremum if and only if the set B has an infimum. Show that when these exist, the supremum of A need not always coincide with the infimum of B. Give two examples of a cut (A, B) where A does not have a supremum: one where exactly one of A and B is empty, and another where none of A and B are empty. Open cuts and topological disconnectedness SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness Cuts (A, B) where A or B are open Show that in a cut (A, B) of a linearly ordered set (L, 6), where neither A nor B are empty, A is open in the open interval topology exactly when the following holds: if A contains an element which is the supremum of A then the infimum of B is contained in B. Formulate the dual result. Deduce a characterization of those cuts (A, B) where both A and B are open. Theorem A linearly ordered set (L, 6) gives rise to a disconnected space via the topology of open intervals if and only if either L is empty, or L admits a cut (A, B) where both A and B are open and none of A and B are empty. Sketch of the proof SAM III General Topology Lecture 3 Contents Connectedness Discrete and indiscrete spaces Definition Some properties Dedekind cuts Definition and basic properties Open cuts and topological disconnectedness The “if part” of the theorem is obvious. We sketch the proof of the “only if” part. Given a decomposition L = X ∪ Y where X and Y are nonempty disjoint open sets, we pick x ∈ X and y ∈ Y and assume, without loss of generality, that x < y . Then, we construct a cut (A, B) as follows: first, define U to be the union of all open intervals I such that x ∈ I ⊆ X ; then, take A to be the smallest down-closed set whose subset is U (i.e. A consists of all a ∈ L such that ∃u∈U a 6 u); finally, take B = L \ A. It then turns out that A is open, x ∈ A and y ∈ B. If we assume that B is not open, then we will get that B contains its infimum b which is at the same time the supremum of A. It is then verified that assuming b ∈ X and b ∈ Y both lead to a contradiction. Since L = X ∪ Y this shows that B must be open, concluding the proof.