Flashcards
§6 Connectedness
Cards in this deck (24)
- 1Definition 6.1: Separation, Disconnected & ConnectedA \(\underline{separation}\) of a space X is a pair U,V of disjoint, non-empty, open subsets whose union is X:
- U,V open
- U ≠ ∅ ≠ V
-U∩V = ∅
- U∪V = X
X is \(\underline{disconnected}\) if there is a separation of X.
X is \(\underline{connected}\) if the is NO separation of X. - 2Fact: X is connected iff...X is connected iff there is no pair of disjoint non-empty \(\underline{closed}\) sets whose union is X.
- 3Theorem 6.2X is \(\underline{connected}\) iff there is no non-empty proper subset 0f X that is clopen.
- 4DefinitionSubsets A & B of space X are \(\underline{separated}\) (from each other) if
A∩Cl(B)= ∅ &
B∩Cl(A)= ∅ - 5TheoremX is connected iff it is not the union of two non-empty separated sets.
- 6Definition 6.2: connected in space XA subset A is \(\underline{connected-in-X}\) if A is a connected space under the subspace topology inherited from X. Otherwise A is \(\underline{disconnected-in-X}\).
- 7Theorem 6.4A is disconnected in X iff there are open subsets U & V of X s.t.
U∩A ≠ ∅ &
V∩A ≠ ∅,
U∩V∩A= ∅ & A⊆U∪V
This means that U∩A & V∩A have union A, i.e. (U∩A)∪(V∩A)=A
& U∩A and V∩A form a separation of A in the subspace. - 8Theorem 6.6If f:X⟶Y is cts & X is connected, then f(x) is connected in Y.
Hence if f is surjective, then Y is connected.
i.e. A cts image of a connected set is connected. - 9Corollary (after Thm 6.6)Connectedness is a topological property, i.e. is preserved by homeomorphism.
- 10Definition: ComponentsAny space is partitioned into connected pieces that are \(\underline{maximally}\) connected. These are called the \(\underline{components}\) of the space. They are closed.
- 11Lemma 6.7Let \(\underline{C}\) be the connected subset of space X & C⊆D⊆X.
If a pair U,V form a separation of D in X, then either C⊆U or C⊆V. - 12Theorem 6.9A collection of connected sets with non-empty intersection has a connected union, i.e. if {\(C_i\): i∈I} has each \(C_i\) connected AND
\(\underset {i∈I}{∩}\)\(C_i\) ≠ ∅
Then (\(\underset {i∈I}{∪}\)\(C_i\)) is connected. - 13Definition 6.11: The Component PartitionFix a space X. Let x,y∈X.
Define x\({~}_c\)y iff there is a connected set C⊆X with x,y∈C.
This is an equivalence relation on X - 14Theorem 6.12: Results(1) Each component \(\underline{is}\) connected.
(11) Each component C is maximally connected, i.e. id C⊂D (a proper subset), then D is not connected.
(111) Any connected subset of the space X is itself a subset of a component. - 15Theorem 6.8If C is a connected subset of space X & C⊆A⊆Cl(C), then A is connected.
- 16Corollary 1 (following Thm 6.8)Putting A=Cl(C), we get:
if C is connected, then Cl(C) is connected. - 17Corollary 2 (following Thm 6.8)If C is a component, C is closed.
- 18Theorem 6.10A product of connected spaces is connected, i.e. if X & Y are connected, so is X×Y.
- 19Definition 6.19: CutsetsA subset S of X is a \(\underline{cutset}\) of space X if X-S is disconnected.
A point p is a \(\underline{cutpoint}\) of X if {p} is a cutset of X. - 20Theorem 6.20Cutsets are preserved by homeomorphism:
if f:X≅Y & S is a cutset of X, then its direct image f(S) is a cutset of Y. - 21Jordan Curve Theorem (Theorem 11.2)Every simple closed curve C in \(ℝ^2_{std}\) is a cutset if the plane with \(ℝ^2\)-S having 2 components & S as boundary.
(simple closed curve = embedding of \(S^1\) in \(ℝ^2_{std}\)) - 22Information: Path ConnectednessA path in a space X is a cts function
p:[0,1\(]_{std}\)⟶X.
The path \(\underline{begins}\) at p(0) in X and \(\underline{ends}\) at p(1).
p need not be injective i.e. path can cross itself.
We identify the oath p with its image p([0.1])⊆X. - 23Defintion 6.27: Path-connectedA space X is \(\underline{path-connected}\) if, ∀x,y∈X there is a path p in X from x to y.
i.e. p(0)=x & p(1)=y - 24Theorem 6.8If X is path-connected than X is connected.
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- Diploma/Certificado de qualificação profissional de nivel 3 Mathematics (Topology (MATH313)) Flashcards on §6 Connectedness, created by Amy Wogan on 29/04/2016.
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