Lecture 23 - 10/17/2012

Math 5801 General Topology and Knot Theory

Nathan Broaddus

Ohio State University

October 17, 2012

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Course Info

Reading for Friday, October 19 Chapter 3.26, pgs. 163-170

HW 8 for Monday, October 22

I Chapter 2.24: 3, 5a-d, 8a-d, 12a-f (see pg. 66 for required deﬁnitions)

I Chapter 2.25: 1, 2a-c

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Connectedness

I How does path connectedness relate to connectedness?

Proposition 217 (Path connected implies connected) If X is a path connected space then X is connected.

Proof.

I Fix x0 ∈ X

I for each x ∈ X let fx : [0, 1] → X be a path from x0 to x. S I X = x∈X fx ([0, 1]) is a union of connected sets.

I What about the other direction?

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Connectedness

Proposition 218 (Connected does not imply path connected) There is a space which is connected but not path connected.

Example 219 (Ordered square is connected but not path connected)

2 I Let Io = I × I with dictionary order topology. 2 I Io is a linear continuum and hence connected. 2 I Suppose f : [0, 1] → Io is continuous and satisﬁes f (0) = (0, 0) and f (1) = (1, 1) 1 2 −1 1 2 I For each r ∈ [0, 1] the set {r} × ( 3 , 3 ) is open so f ({r} × ( 3 , 3 )) is open in [0, 1].

I Thus it contains some q ∈ Q ∩ [0, 1].

I Thus π1 ◦ f is a surjection from countable Q ∩ [0, 1] to uncountable [0, 1].

I Contradiction. Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Connectedness

Example 220 (Topologist’s sine curve)

2 I Let X ⊂ R be the following subspace

1 U = {(x, sin( x ))|x > 0} V = {(0, y)|0 ≤ y ≤ 1} X = U ∪ V

I U is path connected and hence connected.

I U ⊂ X ⊂ U so X is connected.

I No path can connect (0, y) to (1, sin 1).

I Suppose f : [0, 1] → X is a path from (0, y) to (1, sin 1) with f (t) = (x(t), y(t)).

I Assume x(0) = 0 and x(t) > 0 for t > 0 otherwise restrict f to smaller domain. n I Choose (tn)n∈Z+ to be a sequence in I with y(tn) = (−1) and tn → 0

I f cannot be continuous.

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Deﬁnition 221 (Components of a space)

Let X be topological space. For each x, y ∈ X set x ∼c y if X has a connected subset A with x, y ∈ A.A component of X is an equivalence class of ∼c .

I Why is ∼c transitive?

Deﬁnition 222 (Path components of a space)

Let X be topological space. For each x, y ∈ X set x ∼p y if X if there is a path in X from x to y.A path component of X is an equivalence class of ∼p.

I Why is ∼p transitive?

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Deﬁnition 223 (Concatenation of paths) Let f , g : I → X be paths in X such that f (1) = g(0). The concatenation of f and g is the path f · g : I → X where

1 f (2t), 0 ≤ t ≤ 2 f · g(t) = 1 g(2t − 1), 2 ≤ t ≤ 1

Proposition 224 (Path components of a space) If f , g : I → X are paths in X such that f (1) = g(0) then f · g is a path in X from f (0) to g(1).

Proof.

I f cont. and t 7→ 2t cont. so t 7→ f (2t) cont.

I g cont. and t 7→ 2t − 1 cont. so t 7→ g(2t − 1) cont.

I Gluing lemma gives f · g cont. Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Examples 225 (Components and path components)

1. Components of Q are single points.

I Suppose a√, b ∈ A ⊂ Q√and a 6= b. 2 2 I Let m = 2 a + (1 − 2 )b I Let U = {x ∈ A|x < m} and V = {x ∈ A|x > m}.

I Then U and V give a separation of A. 2. Path components of Q are single points.

I Each path component of a space X must be contained in a component of X

I Single points are path connected. 3. Components of topologists’s sine curve X from Example 220 are the space X since X is connected. 4. Path components of topologists’s sine curve X are the space are the sets U and V from Example 220.

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Deﬁnition 226 (Locally connected) A topological space X is locally connected if for all x ∈ X and every open neighborhood Ux of x there is a connected neighborhood Vx of x with Vx ⊂ Ux .

Deﬁnition 227 (Locally path connected) A topological space X is locally path connected if for all x ∈ X and every open neighborhood Ux of x there is a path connected neighborhood Vx of x with Vx ⊂ Ux .

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Examples 228 (Local connectivity)

1. Q is not locally connected or locally path connected. h i 2. S 1 , 1 ⊂ R is locally path connected. n∈Z+ 2n+1 2n h i 3. {0} ∪ S 1 , 1 ⊂ R is not locally path connected. n∈Z+ 2n+1 2n 4. Topologist’s sine curve is not locally connected (hence not locally path connected). 5. R − Q is not locally connected or locally path connected. 6. R2 − Q2 is locally path connected.

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Proposition 229 A space X is locally connected if and only if for every open subset U ⊂ X each component of U is open in X .

Proof.

I Suppose X is locally connected.

I Let U ⊂ X be open.

I Let C ⊂ U be a component of U.

I Let x ∈ C.

I x has a connected open nbdh Vx ⊂ U. I By deﬁnition of component Vx ⊂ C. S I Thus C = x∈C Vx is open. I Suppose for every open subset U ⊂ X each component of U is open in X .

I Let U ⊂ X be open.

I Let C ⊂ U be the component of x ∈ U.

I C is a connected open neighborhood of x contained in U.

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Components

Proposition 230 A space X is locally path connected if and only if for every open subset U ⊂ X each path component of U is open in X .

Proposition 231 If X is locally path connected then its components are also path components.

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Compactness

I Intermediate Value Theorem for R lead us to notion of connectedess.

I Another classic theorem for continuous functions on R is the Maximum Theorem:

If f :[a, b] → R is continuous then there is y ∈ [a, b] s.t. for all x ∈ [a, b] we have f (y) ≥ f (x).

I What topological property of [a, b] ensures that continuous images achieve a maximum value?

I We want a property P such that if X has property P then for any continuous f : X → R there is y ∈ X such that ∀x ∈ X , f (y) ≥ f (x).

I Note that this property is not shared by R or (a, b).

I On the other hand it should (probably) be shared by any ﬁnite set with the discrete topology.

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Compactness

Deﬁnition 232 (Cover) A cover of a topological space X is a collection A of subsets of X such that S A = X .

Deﬁnition 233 (Subcover) A subcover of a cover A of a topological space X is a collection S ⊂ A such that S S = X .

Deﬁnition 234 (Open cover) An open cover of a topological space X is a collection U of open sets such that S U = X .

Deﬁnition 235 (Compact) A topological space X is compact if every open cover of X has a ﬁnite subcover.

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Compactness

Examples 236 (Noncompact Spaces) Showing that X is not compact only requires an inﬁnite open cover of X which has no ﬁnite subcover: 1. R is not compact. 4 I For each n ∈ Z let Un = (n, n + 3 ). I Let U = {Un|n ∈ Z}. S I U is an open cover of R since U = R.

I Let S ⊂ U be a subcover.

I For all n ∈ Z n + 1 ∈ Un and n + 1 ∈/ Um for m 6= n I Hence for each n ∈ Z we have Un ∈ S.

I Thus the only subcover of U is U which is inﬁnite. 2. (0, 1) is not compact. 1 1 I For each n ∈ Z+ let Un = ( 4 , n ). n+ 3 I Let U = {Un|n ∈ Z+}. 3. Q and Z are not compact. 4. Rn is not compact.

Nathan Broaddus General Topology and Knot Theory

Lecture 23 - 10/17/2012 Connectedness Components Compactness Compactness

Deﬁnition 237 (Cover of a subspace) If A is a subspace of X a collection of set S covers A if A ⊂ S S.

Proposition 238 A subspace A ⊂ X is compact if and only if any covering of A by open sets of X has a ﬁnite subcovering.

Nathan Broaddus General Topology and Knot Theory Lecture 23 - 10/17/2012 Connectedness Components Compactness Compactness

Proposition 239 (Continuous images of compact spaces are compact) If X and Y are spaces and X is compact and f : X → Y is continuous then f (X ) is a compact subspace of Y .

Proof.

I Suppose X is compact and f : X → Y is continuous

I Let V be a cover of f (X ) by open sets in Y . −1 I Let U = {f (V )|V ∈ V}.

I U is an open cover of X so it has a ﬁnite subcover −1 −1 {f (V1), ··· , f (Vn)}

I So {V1, ··· , Vn} is a ﬁnite subcover of f (X ).

Nathan Broaddus General Topology and Knot Theory