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# 数学代写|数学分析作业代写Mathematical Analysis代考|MAJ01156 Bases and Subbases

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## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Bases and Subbases

Some topologies are quite difficult to define directly, and it is frequently the case that we want to define a topology on a set $X$ that includes a certain collection S of subsets of $X$. The existence of such a topology is obvious because $\mathcal{P}(X)$ is such a topology. However, $\mathcal{P}(X)$ is useless because it is too large. This immediately suggests the question of finding the smallest topology $\mathcal{J}$ on $X$ that contains $\widetilde{\subseteq}$. Fortunately, such a unique smallest topology $\mathcal{T}$ exists.

The reader may wonder what situations would compel us to “want” the members of $\mathfrak{S}$ to be open. The prime such situation is when we need a certain class of functions from $X$ to another topological space $Y$ to be continuous, which is the overarching idea behind the definition of product and weak topologies. See sections $5.4$ and 6.7.

The set $\Im$ in the above discussion is called a subbase for $\mathcal{T}$, and a closely connected concept is that of a base for the topology $\mathcal{T}$, which is our first definition. Bases and subbases have a wide range of applications. In addition to providing the means to define useful topologies, bases and subbases give us easy ways to prove the continuity of functions and to characterize closures. See theorems 5.2.2 and 5.3.1.
Definition. An open base for a topology $\mathcal{T}$ on a set $X$ is a collection $\mathfrak{B}$ of open subsets of $X$ such that every nonempty open subset in $X$ is the union of members of $\mathfrak{B}$. If $\mathfrak{B}$ is an open base for $\mathcal{T}$, we say that $\mathfrak{B}$ generates $\mathcal{T}$.

See problem 2 at the end of this section for an equivalent, more explicit formulation of the definition of an open base.

Example 1. The collection $\mathfrak{B}={(r, s): r, s \in \mathbb{Q}, r<s}$ is an open base for the usual topology on $\mathbb{R}$. This is because every open subset of $\mathbb{R}$ is the union of open bounded intervals, and any such interval is the union of members of $\mathfrak{B}:(a, b)=$ $\cup{(r, s): r \in \mathbb{Q}, s \in \mathbb{Q}, a<r<s<b}$. See section $4.5$ for a more general version of this example.

The collection of open balls in a metric space is an open base for the metric topology. This follows immediately from the definition of open sets in a metric space.

Caution: Not every collection $\mathfrak{S}$ of subsets of $X$ such that $\cup{U: U \in \mathfrak{E}}=X$ is the open base for some topology on $X$, as the next example illustrates.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Continuity

In section 4.3, we studied the definition of local continuity of functions on metric spaces. It is clear that the $\epsilon-\delta$ definition provides no clues to generalizing the definition to the topological case. However, theorem 4.3.1 provides a metric-free characterization of local continuity which, with very slight changes, produces the following definition.

Definition. Let $X$ and $Y$ be topological spaces. A function $f: X \rightarrow Y$ is said to be continuous at a point $x_{0} \in X$ if, for every open subset $V$ of $Y$ containing $f\left(x_{0}\right)$, $f^{-1}(V)$ contains an open neighborhood of $x_{0}$.

We point out here an important distinction between metric and general topologies. Theorem 4.3.2 established the fact that, in the metric case, continuity is equivalent to sequential continuity. This is not the case for a general topological space. See problem 11 at the end of this section.

As in the metric case, we can define a function from a topological space $X$ to another space $Y$ to be continuous if it is continuous at each point of $X$. However, theorem 4.3.3 suggests a more convenient, and widely used, definition of global continuity.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Bases and Subbases

5.3.1。

$\mathcal{T}$ ，我们说B生成 $\mathcal{T}$.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Continuity

$f^{-1}(V)$ 包含一个开放的邻域 $x_{0}$.

$4.3 .3$ 提出了一个更方便、更广泛使用的全局连续性定义。

## MATLAB代写

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