answer:To compute the fundamental group of the resulting space, we can use the Seifert-van Kampen theorem. First, we need to divide the square into two open sets U and V such that their intersection is simply connected. We can choose U to be the square with a small open disk removed around the center point x, and V to be an open disk centered at x that covers the edges of the square. Both U and V are simply connected, and their intersection U ∩ V is also simply connected. According to the Seifert-van Kampen theorem, the fundamental group of the resulting space is the free product of the fundamental groups of U and V, divided by the normal subgroup generated by the relations between the fundamental groups of U and V. Since U, V, and U ∩ V are all simply connected, their fundamental groups are trivial (i.e., the group with only one element). Therefore, the fundamental group of the resulting space is the free product of two trivial groups, which is also a trivial group. So, the fundamental group of the resulting space is the trivial group, denoted as {1} or {e}, where e is the identity element.

answer:To compute the fundamental group π₁(X, a) for some point a in the interior of A, we can use the Seifert-van Kampen theorem. The theorem states that if X is the union of two open, path-connected sets U and V such that their intersection U ∩ V is also path-connected, then the fundamental group of X is given by the free product of the fundamental groups of U and V, modulo the normal subgroup generated by the elements of the form i_U(γ) * i_V(γ)^{-1}, where γ is an element of π₁(U ∩ V, a) and i_U and i_V are the inclusion maps. Let U be the interior of the square X, and let V be the square X with the bottom edge removed. Then, U and V are both open and path-connected, and their intersection U ∩ V is also path-connected. The fundamental group of U is trivial, as U is contractible, and the fundamental group of V is isomorphic to the integers, as V is homotopy equivalent to a circle. Now, we need to compute the normal subgroup generated by the elements of the form i_U(γ) * i_V(γ)^{-1}. Since the fundamental group of U is trivial, i_U(γ) is the identity for all γ in π₁(U ∩ V, a). Therefore, the normal subgroup is generated by the elements of the form i_V(γ)^{-1}, which are just the inverses of the elements of π₁(V, a). Since π₁(V, a) is isomorphic to the integers, its elements are just integer multiples of a generator, say g. The inverses of these elements are the integer multiples of the inverse of g, say g^{-1}. Thus, the normal subgroup is generated by the elements of the form g^{-n} for n in the integers. The fundamental group of X is then given by the free product of the fundamental groups of U and V, modulo this normal subgroup. Since the fundamental group of U is trivial, the free product is just π₁(V, a), which is isomorphic to the integers. Modulo the normal subgroup generated by the elements of the form g^{-n}, the fundamental group of X is still isomorphic to the integers. Therefore, the fundamental group π₁(X, a) is isomorphic to the integers, denoted as Z.

answer:To compute the fundamental group of the circle S^1 at the basepoint x_0 = (1,0), we will use the loop representation of the fundamental group. Recall that the fundamental group pi_1(S^1, x_0) is the set of homotopy classes of loops in S^1 based at x_0, with the group operation given by concatenation of loops. Consider a loop gamma: [0,1] to S^1 based at x_0. Since S^1 is the image of the continuous map f(t) = (cos(2pi t), sin(2pi t)), we can represent gamma as a continuous map tilde{gamma}: [0,1] to [0,1] such that gamma(t) = f(tilde{gamma}(t)) for all t in [0,1]. Note that tilde{gamma}(0) = tilde{gamma}(1) = 0 since gamma is a loop based at x_0. Now, consider two loops gamma_1 and gamma_2 in S^1 based at x_0, with corresponding continuous maps tilde{gamma}_1 and tilde{gamma}_2. The concatenation of these loops, denoted by gamma_1 * gamma_2, can be represented by the continuous map tilde{gamma}_1 * tilde{gamma}_2: [0,1] to [0,1] defined as follows: (tilde{gamma}_1 * tilde{gamma}_2)(t) = begin{cases} 2tilde{gamma}_1(t) & text{if } 0 leq t leq frac{1}{2}, 2tilde{gamma}_2(t) - 1 & text{if } frac{1}{2} < t leq 1. end{cases} This map is continuous and satisfies (tilde{gamma}_1 * tilde{gamma}_2)(0) = (tilde{gamma}_1 * tilde{gamma}_2)(1) = 0. Thus, the concatenation of loops in S^1 corresponds to the concatenation of continuous maps from [0,1] to itself with the same endpoints. Now, we claim that the fundamental group pi_1(S^1, x_0) is isomorphic to the group of integers mathbb{Z} under addition. To see this, consider the winding number of a loop gamma around x_0. The winding number is an integer that counts the number of times gamma winds around x_0 in the counterclockwise direction, with clockwise windings counted as negative. Given a loop gamma with winding number n, we can construct a homotopy between gamma and the loop that winds around x_0 exactly n times. This shows that the winding number is a homotopy invariant, and thus, it induces a group homomorphism w: pi_1(S^1, x_0) to mathbb{Z}. Moreover, the winding number homomorphism is surjective, as for any integer n, we can construct a loop gamma_n that winds around x_0 exactly n times. Finally, the winding number homomorphism is injective, as two loops with the same winding number are homotopic. Therefore, the winding number homomorphism is an isomorphism between pi_1(S^1, x_0) and mathbb{Z}. In conclusion, the fundamental group of the circle S^1 at the basepoint x_0 = (1,0) is isomorphic to the group of integers mathbb{Z} under addition, i.e., pi_1(S^1, x_0) cong mathbb{Z}.

answer:To compute the fundamental group of the space obtained by removing the point (2,6) from the circle, we will use the concept of homotopy and homotopy groups. Let's denote the circle with radius 5 centered at the point (2,3) as C and the point (2,6) as P. The space obtained by removing the point P from the circle C is denoted as X, i.e., X = C - {P}. Now, we want to compute the fundamental group of X, denoted as π_1(X). First, we observe that X is path-connected, meaning that for any two points in X, there exists a continuous path connecting them. This is because we have only removed one point from the circle, and the remaining space is still connected. Next, we will use the fact that the fundamental group of a path-connected space is independent of the choice of basepoint. So, we can choose any point on X as our basepoint. Let's choose the point (2,3) as our basepoint, denoted as x_0. Now, we want to find the set of homotopy classes of loops in X based at x_0. A loop in X is a continuous map f: [0,1] → X such that f(0) = f(1) = x_0. We can observe that any loop in X can be continuously deformed to a loop that goes around the missing point P a certain number of times. This number of times can be positive, negative, or zero, depending on the direction of the loop. This means that the homotopy classes of loops in X can be represented by integers, where the integer n represents a loop that goes around the missing point P n times. Now, we can define a group operation on these integers, which is just the usual addition of integers. This operation is associative, has an identity element (0), and has inverses for each element (the inverse of n is -n). Thus, the fundamental group of X, π_1(X), is isomorphic to the group of integers under addition, denoted as ℤ. So, π_1(X) ≅ ℤ.