Connectedness in Topology

Connected Space > s.a. graph; lie hroup representations.
* Idea: A space which is "all in one piece"; Of course, this depends crucially on the topology imposed on the set; Every discrete topological space is "totally" disconnected.
$ Def: A topological space (X, τ) is connected if the only subsets which are both open and closed are Ø and X.
$ Alternatively: (X, τ) is connected if there are no non-trivial U, V ∈ τ such that U ∪ V = X and U ∩ V = Ø.
$ Locally connected space: For all x ∈ X and neighborhoods U of x, there is another neighborhood V ⊂ U, with V connected.
* Extremely disconnected space: One in which every open set has a closure which is open.
* Totally disconnected space: One in which each connected component is a single point; The only perfect, totally disconnected metric topological space is the Cantor set, a fractal.

Arcwise (or Pathwise) Connected Space
$ Def: A topological space (X, τ) is arcwise connected if for all a, b ∈ X there is a continuous path q: [0,1] → X, with q(0) = a and q(1) = b.
* Relationships: Arcwise connectedness implies connectedness in the usual sense, but not viceversa; A counterexample is X ∪ Y ⊂ \(\mathbb R\)², with X = {(0, x₂) | −1 < x₂ < 1}, Y = {(x₁, sin(π/x₁) | 0 < x₂ < 1}.
$ Locally: (X, τ) is locally arcwise connected if for all x ∈ X, and any neighborhood V(x), ∃ U(x) ⊂ V(x), such that U(x) is arcwise connected.
* Relationships: Local arcwise connectedness implies local connectedness; There are topological spaces which are simply connected, but not locally pathwise connected, or not locally connected (think of comb spaces).
@ References: in Singer & Thorpe 67, ch III.

Simply and Multiply Connected Space
$ Simply connected: A pathwise connected space X with trivial fundamental group, π₁(X) = {0}.
$ m-connected: A space X with π_p(X) = {0} for 0 ≤ p ≤ m.
$ Semi-locally simply connected: A space (X,τ) such that for all x ∈ X there is a neighborhood U of x such that any loop in U based at x can be shrunk to a point in X (not necessarily in U).
* Counterexample: X = ∪_n=1^∞ C_n, where C_n is a circle in \(\mathbb R\)² with center at (1/n, 0) and radius 1/n.

Connected Sum of Manifolds or Topological Spaces > s.a. 3D manifolds; laplacian.
$ Def: In sloppy notation, X # Y:= (X \ Dⁿ) ∪ (Y \ Dⁿ), where n is the dimension of X and Y.
* Properties: Associative and commutative; The identity is Sⁿ.
* Examples: X # \(\mathbb R\)ⁿ = X \ {p}.

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