1. Topology is also used for analyzing spatial relationships in many situations, such as dissolving the boundaries between adjacent polygons with the same attribute values or traversing a network of the elements in a topology graph. Since U is an open set, it is the union of $$B_i,\ i\in I$$, where $$B_i\in\mathcal{B}$$ and $$I$$ is some index set. $$\mathcal{T}_1$$ is the set of all possible unions of members of $$\mathcal{B}$$). Definition A set of subsets is a basis of a topology if every open set in is a union of sets of . Hence the topology satisfies the second axiom of countability. F is still countable because it is isomorphic to $$\mathbb{Z}\times\mathbb{Z}$$, which is the Cartesian product of two countable sets. You will find one in a proof somewhere in this post! Example 1. 2. In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. An n-ball (the surface of the n-sphere and the space within it) is a closed set in $$\mathbb{R}^{n+1}$$. (i.e. ( a, b) ⊂ ℝ. Let’s call $$\mathcal{T}_1$$ the topology generated by $$\mathcal{B}$$ (i.e. Basis for a Topology 4 4. And by definition of a basis, we conclude $$\mathcal{B}$$ is a basis of the finite-closed topo. $$\mathcal{B}_2$$ is a basis of $$\mathcal{T}_1$$), $$\forall x,B:x\in B\in\mathcal{B}_2$$, there exists $$B'\in\mathcal{B}_1$$ such that $$x\in B'\subseteq B$$. If X and Y are topological spaces, then the corresponding topology on X × Y is defined by the basis Let $$S=\{0, 1, 1/2, 1/3, \dots, 1/n,\dots\}$$. ♦ So let’s generalise the definition of product space to arbitrary topological spaces. In symbols: if is a set, a collection of subsets of is said to form a basis for a topology on if the following two conditions are satisfied: The second condition is sometimes stated as follows: if , then there exists such that . (This is another example of a countable set not being a closed set.). Then $$\mathcal{T}_1=\mathcal{T}_2$$ iff: Definition. Theorem. If and , then there is a basis element containing such that .. Bases, subbases for a topology. There are certains conditions so that $$\mathcal{B}$$ is a basis. See the 2 2. The collection of all open intervals $$(a, b)$$ whereas $$a,b\in\mathbb{Q}$$ is the basis of $$\mathbb{R}$$. By the triangle inequality, we have $$d$$ smaller than the total distance from the center to the point $$(r_x, r_y)$$ and the distance from there to the furthest vertex. 2. Theorem. 1. We see that the finite intersection of some sets $$X-\{x_i\},\ i\in\{1,\dots,n\}$$, $$x_i\in X$$ is $$X-\{x_1,x_2,\dots,x_n\}$$, which is an open set of the finite-closed topology on X. Every disc \(\{(x,y):(x-a)^2+(y-b)^2