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| Question | Answer |
| Definition 6.1: Separation, Disconnected & Connected | A \(\underline{separation}\) 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. |
| Fact: X is connected iff... | X is connected iff there is no pair of disjoint non-empty \(\underline{closed}\) sets whose union is X. |
| Theorem 6.2 | X is \(\underline{connected}\) iff there is no non-empty proper subset 0f X that is clopen. |
| Definition | Subsets A & B of space X are \(\underline{separated}\) (from each other) if A∩Cl(B)= ∅ & B∩Cl(A)= ∅ |
| Theorem | X is connected iff it is not the union of two non-empty separated sets. |
| Definition 6.2: connected in space X | A subset A is \(\underline{connected-in-X}\) if A is a connected space under the subspace topology inherited from X. Otherwise A is \(\underline{disconnected-in-X}\). |
| Theorem 6.4 | A 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. |
| Theorem 6.6 | If 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. |
| Corollary (after Thm 6.6) | Connectedness is a topological property, i.e. is preserved by homeomorphism. |
| Definition: Components | Any space is partitioned into connected pieces that are \(\underline{maximally}\) connected. These are called the \(\underline{components}\) of the space. They are closed. |
| Lemma 6.7 | Let \(\underline{C}\) 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. |
| Theorem 6.9 | A 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. |
| Definition 6.11: The Component Partition | Fix 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 |
| Theorem 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. |
| Theorem 6.8 | If C is a connected subset of space X & C⊆A⊆Cl(C), then A is connected. |
| Corollary 1 (following Thm 6.8) | Putting A=Cl(C), we get: if C is connected, then Cl(C) is connected. |
| Corollary 2 (following Thm 6.8) | If C is a component, C is closed. |
| Theorem 6.10 | A product of connected spaces is connected, i.e. if X & Y are connected, so is X×Y. |
| Definition 6.19: Cutsets | A 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. |
| Theorem 6.20 | Cutsets 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. |
| Jordan 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 \(ℝ^2_{std}\)) |
| Information: Path Connectedness | A 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. |
| Defintion 6.27: Path-connected | A 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 |
| Theorem 6.8 | If X is path-connected than X is connected. |
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