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The Topology of Path Component Spaces The topology of path component spaces Jeremy Brazas October 26, 2012 Abstract The path component space of a topological space X is the quotient space of X whose points are the path components of X. This paper contains a general study of the topological properties of path component spaces including their relationship to the zeroth dimensional shape group. Path component spaces are simple-to-describe and well-known objects but only recently have recieved more attention. This is primarily due to increased qtop interest and application of the quasitopological fundamental group π1 (X; x0) of a space X with basepoint x0 and its variants; See e.g. [2, 3, 5, 7, 8, 9]. Recall qtop π1 (X; x0) is the path component space of the space Ω(X; x0) of loops based at x0 X with the compact-open topology. The author claims little originality here but2 knows of no general treatment of path component spaces. 1 Path component spaces Definition 1. The path component space of a topological space X, is the quotient space π0(X) obtained by identifying each path component of X to a point. If x X, let [x] denote the path component of x in X. Let qX : X π0(X), q(x) = [x2] denote the canonical quotient map. We also write [A] for the! image qX(A) of a set A X. Note that if f : X Y is a map, then f ([x]) [ f (x)]. Thus f determines a well-defined⊂ function f !: π (X) π (Y) given by⊆f ([x]) = [ f (x)]. 0 0 ! 0 0 Moreover, f0 is continuous since f0qX = qY f and qX is quotient. Altogether, we obtain an endofunctor of the category Top of topological spaces. Proposition 2. π : Top Top is a functor. 0 ! Remark 3. If X has a given basepoint x0, we take the basepoint of π0(X) to be [x0] and denote the resulting based space as π0(X; x0). This gives a functor on the category of based topological spaces. Definition 4. A space X is semi-locally 0--connected if for every point x X, there is a neighborhood U of x such that the inclusion i : U X induces2 the constant map i : π (U) π (X). ! 0 0 ! 0 1 Note that π0(X) is discrete if and only if each path component of X is open. Proposition 5. π0(X) is discrete iff X is semi-locally 0-connected. Proof. If X is semi-locally 0-connected and x X, then there is a neighborhood 2 U of x such that the inclusion i : U X induces the constant map i0 : π0(U) π (X). Thus U [x] and [x] is open! in X. Since any given path component! of 0 ⊆ X is open, π0(X) is discrete. Conversely, suppose π0(X) is discrete. If x X, let U be the open set [x]. Clearly the inclusion [x] X induces the constant2 map π ([x]) π (X). ! 0 ! 0 Corollary 6. If X is locally path connected, then π0(X) is discrete. Example 7. If X is a geometric simplicial complex or a CW-complex, then π0(X) is discrete. Another case of interest is when the natural quotient map qX : X π0(X) is a homeomorphism. ! Remark 8. The quotient map qX : X π (X) is a homeomorphism iff it is ! 0 injective. Thus qX is a homeomorphism iff X is totally path disconnected (i.e. each path component consists of a single point). Specifically, X π0(X) if X is totally disconnected or zero dimensional (there is a basis for the topology of X consisting of clopen sets). Another general and useful fact about path component spaces is that every space is a path component space. Theorem 9. [10] For every space X, there is a paracompact Hausdorff space H(X) and a natural homeomorphism π0(H(X)) X. 2 Examples Example 10. Let T R2 be the topologist’s sine curve ⊆ 1 T = (0; 0) (x; y) y = sin ; 0 < x 1/π f g [ j x ≤ and Tc = T 0 [ 1; 1] be the closed topologist’s sine curve. It is easy to see [ f g × − that in both cases, the path component space π0(T) π0(Tc) is homeomorphic to the Sierpinski space S = 0; 1 with topology ; 1 ; 0; 1 . In particular, the open set 1 cooresponds tof theg path componentf; off (1g/π,f 0).gg f g The above example allows us to characterize when certain path component spaces are T1. Recall that a space X is sequential if for every non-closed set A X, there is a convergent sequence xn x such that xn A and x < A. It is⊂ well-known that metric spaces are sequential! and sequential2 spaces are precisely those which are quotients of metric spaces. 2 Proposition 11. If π0(X) is T1, then any map f : T X from the topologist’s sine curve induces the constant map f : S π (X). If X is! sequential, the converse holds. 0 ! 0 Proof. Let A = (0; 0) and B = T A be the path components so that π0(T) = f g n 1 A; B with topology ; B ; A; B and let tn = nπ , n 1. Suppose π0(X) is T1 f g f; f g f gg ≥ 1 and f : T X be a map. Since the singleton f (B) is closed in X, f − ( f (B)) ! f g 0 is non-empty and closed in π0(T). Since the only non-empty closed set in 1 π (T) containing B is π (T) = A; B , we have f − ( f (B)) = A; B . Therefore 0 0 f g 0 f g f0(A) = f0(B) showing f0 is constant. Suppose X is sequential and x X. To obtain a contradiction, suppose the 2 singleton [x] is not closed in π (X). Since qX : X π (X) is quotient, [x] is not f g 0 ! 0 closed in X. Thus there is a convergent sequence xn x such that xn [x] and ! 2 x < [x]. Define a function f : T X by f (tn) = xn and f (0; 0) = x. Extend f to ! the rest of T by defining f on the arc from tn to tn+1 to be a path from xn to xn+1 in X (which exists since xn [x] for all n). Clearly f is continuous and induces an injection f : S π (X).2 0 ! 0 One can generalize this approach to general spaces by replacing convergent sequences with convergent nets and T by analogous constructions based on directed sets. The next example allows us to realiize subspaces of R as path component spaces of simple spaces constructed independently of Theorem 9. Example 12. Let X be the set R [0; 1]. We define a Hausdorff topology × on X such that π0(A [0; 1]) A for each subset of the form A [0; 1] X. The topology on X has× a basis consisting of sets of the form a× (s; t)⊆ and a (t; 1] (a; b) [0; 1] b [0; s) for 0 < s < t < 1 and a < b.f Thisg × topology isf g a × simple[ extension× of[ the f gordered × square in [14, 16, Example 3] and is the order topology given by the dictionary ordering onx X. The path components of X are z [0; 1] for z R (see [14, 24, Example 6]). It then suffices to f g × 2 x show that for each A R, the projection pA : XA A is quotient, where ⊆ ! XA = A [0; 1] has the subspace topology of X. Suppose U is open in R × 1 so that U A is open in A. Since U [0; 1] = pR− (U) is open in X and so \ 1× (U [0; 1]) XA = (U A) [0; 1] = pA− (U A) is open in XA. Therefore pA is × \ \ × \ 1 continuous. Now suppose V A such that p− (V) = V [0; 1] is open in XA. For ⊆ A × each v V, there is an open neighborhood v (tv; 1] (v; bv) I bv [0; sv) 2 f g × [ × [ f g × of (v; 1) contained in V [0; 1]. Since V [0; 1] is saturated with respect to pA, × × we have ([v; bv] A) [0; 1] V [0; 1]. Similarly, since (v; 0) V for each v V \ × ⊂ × 2 2 we can find a closed interval [av; v] such that ([av; v] A) [0; 1] V [0; 1]. \ × ⊂ × Therefore, for each v V, we have v (av; bv) A V. Therefore V is open in 2 2 \ ⊆ A. Thus pA is quotient, and consequently π0(XA) A. Example 13. Using the previous example, we can find a smiple space Y such 1 1 that π0(Y) S . Let : R S denote the exponential map and X = R [0; 1] be the space defined in the! previous example. Let Y be the set S1 [0; 1]× with the quotient topology with respect to id : X S1 [0; 1]. The× projection 1 × ! × 1 Y S is precisely the quotient map qY : Y π (Y) and so S π (Y). ! ! 0 0 3 3 Limits and colimits Proposition 14. π : Top Top preserves coproducts and quotients. 0 ! ` ` Proof. It is immediate that π ( X ) π (X ). If p : X Y is a quotient 0 λ λ λ 0 λ ! map, then qY p = p0 qX is a quotient map. Since qX is quotient, p0 = π0(p) is quotient. ◦ ◦ Though π0 preserves coproducts, unfortunately it fails to be cocontinuous (in the sense that it preserves all colimits).
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