Definition : A topological space X is said to be connected if the empty set ∅ and the whole space X are the only subsets of X that are both open and closed.

Lemma 1.1 A topological space X is connected if and only if it has the following property: if U and V are non-empty open sets in X such that X = U ∪ V , then U ∩ V is nonempty.

Proof If U is a subset of X that is both open and closed, and if V = X U,  then U and V are both open, U ∪ V = X and U ∩ V = ∅. Conversely if U and V are open subsets of X satisfying U ∪ V = X and U ∩ V = ∅, then U = X V , and hence U is both open and closed. Thus a topological space X is connected if and only if there do not exist non-empty open sets U and V such that U ∪ V = X and U ∩ V = ∅. The result follows.

Let Z be the set of integers with the usual topology (i.e., the subspace topology on Z induced by the usual topology on R). Then {n} is open for all n ∈ Z, since

                                    {n} = Z ∩ {t ∈ R : |t − n| <1/2}.

It follows that every subset of Z is open (since it is a union of sets consisting of a single element, and any union of open sets is open). It follows that a function f: X → Z on a topological space X is continuous if and only if f⁻¹ (V ) is open in X for any subset V of Z. We use this fact in the proof of the next theorem.

Proposition 1.2 A topological space X is connected if and only if every continuous function f: X → Z from X to the set Z of integers is constant.

Proof Suppose that X is connected. Let f: X → Z be a continuous function.

Choose n ∈ f(X), and let

                        U = {x ∈ X : f(x) = n}, V = {x ∈ X : f(x) ≠n}.

Then U and V are the preimages of the open subsets {n} and Z {n} of Z, and therefore both U and V are open in X. Moreover U ∩ V = ∅, and X = U ∪ V . It follows that V = X U, and thus U is both open and closed.  Moreover U is non-empty, since n ∈ f(X). It follows from the connectedness of X that U = X, so that f: X → Z is constant, with value n.

Conversely suppose that every continuous function f: X → Z is constant.

Let S be a subset of X which is both open and closed. Let f: X → Z be defined by

Now the preimage of any subset of Z under f is one of the open sets ∅, S, X S and X. Therefore the function f is continuous. But then the function f is constant, so that either S = ∅ or S = X. This shows that X is connected.

Lemma 1.3 The closed interval [a, b] is connected, for all real numbers a and b satisfying a ≤ b.

Proof Let f: [a, b] → Z be a continuous integer-valued function on [a, b]. We show that f is constant on [a, b]. Indeed suppose that f were not constant.  Then f(τ ) ≠f(a) for some τ ∈ [a, b]. But the Intermediate Value Theorem would then ensure that, given any real number c between f(a) and f(τ ), there would exist some t ∈ [a, τ ] for which f(t) = c, and this is clearly impossible,  since f is integer-valued. Thus f must be constant on [a, b]. We now deduce from Proposition 1.2 that [a, b] is connected.

Example: Let X = {(x, y) ∈ R²: x ≠0}. The topological space X is not connected. Indeed if f: X → Z is defined by

then f is continuous on X but is not constant.

A concept closely related to that of connectedness is path-connectedness.

Let x₀ and x₁ be points in a topological space X. A path in X from x₀ to x₁ is defined to be a continuous function γ: [0, 1] → X such that γ(0) = x₀ and γ(1) = x₁. A topological space X is said to be path-connected if and only if,  given any two points x₀ and x₁ of X, there exists a path in X from x₀ to x₁.

Proposition 1.4 Every path-connected topological space is connected.

Proof Let X be a path-connected topological space, and let f: X → Z be a continuous integer-valued function on X. If x₀ and x₁ are any two points of X then there exists a path γ: [0, 1] → X such that γ(0) = x₀ and γ(1) = x₁. But then f ◦ γ: [0, 1] → Z is a continuous integer-valued function on [0, 1]. But  [0, 1] is connected (Lemma 1.3), therefore f◦γ is constant (Proposition 1.2).  It follows that f(x₀) = f(x₁). Thus every continuous integer-valued function on X is constant. Therefore X is connected, by Proposition 1.30.

The topological spaces R, C and Rⁿ are all path-connected. Indeed, given any two points of one of these spaces, the straight line segment joining these two points is a continuous path from one point to the other. Also the n-sphere Sⁿ is path-connected for all n > 0. We conclude that these topological spaces are connected.

Let A be a subset of a topological space X. Using Lemma 1.1 and the definition of the subspace topology, we see that A is connected if and only if the following condition is satisfied:

• if U and V are open sets in X such that A∩U and A∩V are non-empty and A ⊂ U ∪ V then A ∩ U ∩ V is also non-empty.

Lemma 1.5 Let X be a topological space and let A be a connected subset of X. Then the closure Ᾱ of A is connected.

Proof It follows from the definition of the closure of A that Ᾱ ⊂ F for any closed subset F of X for which A ⊂ F. On taking F to be the complement of some open set U, we deduce that Ᾱ∩ U = ∅ for any open set U for which A ∩ U = ∅. Thus if U is an open set in X and if Ᾱ ∩ U is non-empty then A ∩ U must also be non-empty.

Now let U and V be open sets in X such that Ᾱ ∩ U and Ᾱ∩ V are non-empty and Ᾱ ⊂ U ∪ V . Then A ∩ U and A ∩ V are non-empty, and A⊂ U ∪ V . But A is connected. Therefore A ∩ U ∩ V is non-empty, and thus Ᾱ ∩ U ∩ V is non-empty. This shows that Ᾱ is connected.

Lemma 1.6 Let f:X → Y be a continuous function between topological spaces X and Y , and let A be a connected subset of X. Then f(A) is connected.

Proof Let g: f(A) → Z be any continuous integer-valued function on f(A).

Then g ◦ f: A → Z is a continuous integer-valued function on A. It follows from Proposition 1.30 that g ◦ f is constant on A. Therefore g is constant on f(A). We deduce from Proposition 1.30 that f(A) is connected.

Lemma 1.7 The Cartesian product X × Y of connected topological spaces X and Y is itself connected.

Proof Let f: X×Y → Z be a continuous integer-valued function from X×Y to Z. Choose x0 ∈ X and y₀ ∈ Y . The function x → f(x, y0) is continuous on X, and is thus constant. Therefore f(x, y0) = f(x0, y0) for all x ∈ X. Now fix x. The function y → f(x, y) is continuous on Y , and is thus constant.

Therefore

                      f(x, y) = f(x, y₀) = f(x₀, y₀)

for all x ∈ X and y ∈ Y . We deduce from Proposition 1.30 that X × Y is connected.

We deduce immediately that a Finite Cartesian product of connected topological spaces is connected.

Proposition 1.8 Let X be a topological space. For each x ∈ X, let S_x be the union of all connected subsets of X that contain x. Then

(i) S_x is connected,

(ii) S_x is closed,

(iii) if x, y ∈ X, then either S_x = S_y, or else S_x ∩ S_y = ∅.

Proof Let f: S_x→ Z be a continuous integer-valued function on S_x, for some x ∈ X. Let y be any point of S_x. Then, by definition of S_x, there exists some connected set A containing both x and y. But then f is constant on A,  and thus f(x) = f(y). This shows that the function f is constant on S_x. We deduce that S_x is connected. This proves (i). Moreover the closure  is connected, by Lemma 1.33. Therefore v  ⊂ S_x. This shows that S_x is closed,  proving (ii).

Finally, suppose that x and y are points of X for which Sx ∩ Sy ≠∅. Let f: S_x ∪ S_y → Z be any continuous integer-valued function on S_x ∪ S_y. Then f is constant on both S_x and S_y. Moreover the value of f on S_x must agree with that on S_y, since S_x ∩ S_y is non-empty. We deduce that f is constanton S_x ∪ S_y. Thus S_x ∪ S_y is a connected set containing both x and y, and thus S_x ∪S_y ⊂ S_x and S_x ∪S_y ⊂ S_y, by definition of S_x and S_y. We conclude that S_x = S_y. This proves (iii).

Given any topological space X, the connected subsets S_x of X defined as n the statement of Proposition 1.8 are referred to as the connected components of X. We see from Proposition 1.8, part (iii) that the topological space X is the disjoint union of its connected components.

Example : The connected components of

 {(x, y) ∈ R²: x ≠ 0} are{(x, y) ∈ R²: x > 0} and {(x, y) ∈ R²: x < 0}

Example : The connected components of

                  {t ∈ R : |t − n| <1/2for some integer n}.

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