# 数学代写|拓扑学代写Topology代考|THE DEFINITION AND SOME EXAMPLES

## 数学代写|拓扑学代写Topology代考|THE DEFINITION AND SOME EXAMPLES

Let $X$ be a non-empty set. A class $\mathrm{T}$ of subsets of $X$ is called a topology on $X$ if it satisfies the following two conditions:
(1) the union of every class of sets in $T$ is a set in $T$;
(2) the intersection of every finite class of sets in $T$ is a set in $T$.
A topology on $X$ is thus a class of subsets of $X$ which is closed under the formation of arbitrary unions and finite intersections. A topological space consists of two objects: a non-empty set $X$ and a topology $\mathrm{T}$ on $X$. The sets in the class $T$ are called the open sets of the topological space $(X, T)$, and the elements of $X$ are called its points. It is customary to denote the topological space $(X, T)$ by the symbol $X$ which is used for its underlying set of points. No harm can come from this practice if one clearly understands that a topological space is more than merely a non-empty set: it is a non-empty set together with a specific topology on that set. We shall often be considering several topologies on a single given set, and in these circumstances distinct topologies make the set into distinct topological spaces. We observe that the empty set and the full space are always open sets in every topological space, since they are the union and intersection of the empty class of sets, which is a subclass of every topology.

We now list several simple examples of topological spaces. In order to exhibit a topological space, one must specify a non-empty set, tell which subsets are to be considered the open sets, and verify that this given class of sets satisfies conditions (1) and (2) above. In the examples which follow, we leave this third step to the reader.

Example 1. Let $X$ be any metric space, and let the topology be the class of all subsets of $X$ which are open in the sense of the definition in Sec. 10. This is called the usual topology on a metric space, and we say that these sets are the open sets generated by the metric on the space. Metric spaces are the most important topological spaces, and whenever we speak of a metric space as a topological space, it is understood (unless we say something to the contrary) that its topology is the usual topology described here.

## 数学代写|拓扑学代写Topology代考|ELEMENTARY CONCEPTS

We have taken open sets as the starting point in our development of topology, and we now define a number of other basic concepts in terms of open sets. Most of these will be familiar to the reader from the previous chapter, and he will observe that in every case the definition given here is a strict generalization of our earlier definition or some equivalent form of it.

A closed set in a topological space is a set whose complement is open. The following theorem is an immediate consequence of Eqs. 2-(2) and the assumed properties of open sets.

Theorem A. Let $X$ be a topological space. Then (1) any intersection of closed sets in $X$ is closed; and (2) any finite union of closed sets in $X$ is closed.
By considering the empty class of closed sets, we see at once that the empty set and the full space-its union and intersection-are always slosed sets in every topological space.

If $A$ is a subset of a topological space, then its closure (denoted by $\bar{A}$ ) is the intersection of all closed supersets of $A$. It is easy to see that the closure of $A$ is a closed superset of $A$ which is contained in every closed superset of $A$, and that $A$ is closed $\Leftrightarrow A=\bar{A}$. A subset $A$ of a topological space $X$ is said to be dense (or everywhere dense) if $\bar{A}=X$, and $X$ is called a separable space if it has a countable dense subset. For reasons which will become clear at the end of this section, we summarize the main facts about the operation of forming closures in the following theorem. Its proof is a direct application of the above statements.

Theorem B. Let $X$ be a topological space. If $A$ and $B$ are arbitrary subsets of $X$, then the operation of forming closures has the following four properties: (1) $\bar{\emptyset}=\emptyset$; (2) $A \subseteq \bar{A} ;$ (3) $\bar{A}=\bar{A}$; and (4) $\overline{A \cup B}=\bar{A} \cup \bar{B}$.
A neighborhood of a point (or a set) in a topological space is an open set which contains the point (or the set). A class of neighborhoods of a point is called an open base for the point (or an open base at the point) if each neighborhood of the point contains a neighborhood in this class. In the case of a point in a metric space, an open sphere centered on the point is a neighborhood of the point, and the class of all such open spheres is an open base for the point. Our next theorem gives a useful characterization (in terms of neighborhoods) of the closure of a set.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|THE DEFINITION AND SOME EXAMPLES

(1) 中每一类集合的并集吨是一个集合吨;
(2) 中每个有限类集合的交集吨是一个集合吨.

## 数学代写|拓扑学代写Topology代考|ELEMENTARY CONCEPTS

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