Download SAM III General Topology

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.