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# 数学代写|拓扑学代写TOPOLOGY代考|MATH525 THE STONE-WEIERSTRASS THEOREMS

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## 数学代写|拓扑学代写TOPOLOGY代考|THE STONE-WEIERSTRASS THEOREMS

Our previous characterizations of compactness for a metric space strongly suggest that this property is related to completeness and total boundedness in some way yet to be formulated. We begin by proving a theorem which clarifies this situation.

Theorem A. A metric space is compact $\Leftrightarrow$ it is complete and totally bounded. PRooF. Let $X$ be a metric space. The first half of our proof is easy, for if $X$ is compact, then it is totally bounded by Theorem 24-D, and it is complete by Problem 12-2 and the fact that every sequence (and therefore every Cauchy sequence) has a convergent subsequence.

We now assume that $X$ is complete and totally bounded, and we prove that $X$ is compact by showing that every sequence has a convergent subsequence. Since $X$ is complete, it suffices to show that every sequence has a Cauchy subsequence. Consider an arbitrary sequence
$$S_{1}=\left{x_{11}, x_{12}, x_{13}, \ldots\right}$$
with this property. These ideas make it possible for us to state the Weierstrass theorem in the following equivalent forms:
(1) the closed subalgebra of $\mathcal{C}[a, b]$ generated by ${1, x}$ equals $\mathcal{C}[a, b]$
(2) any closed subalgebra of $\mathcal{C}[a, b]$ which contains ${1, x}$ equals $\mathcal{C}[a, b]$.

## 数学代写|拓扑学代写TOPOLOGY代考|LOCALLY COMPACT HAUSDORFF SPACES

In Sec. 23 we defined a locally compact space to be a topological space in which each point has a neighborhood with compact closure. Locally compact spaces of ten arise in the applications of topology to geometry and analysis, and since those which do are almost always Hausdorff spaces, we restrict our attention in this section to locally compact Hausdorff spaces.

The main fact about such a space is that it can be converted into a compact Hausdorff space by suitably adjoining a single point. The reader is perhaps familiar from analysis with the prototype of this process, in which the complex plane $C$ is enlarged by adjoining to it an “ideal point” called the point at infinity and denoted by $\infty$. This ideal point can be thought of as any object not in $C$, and we denote by $C_{\infty}$ the larger set $C \cup{\infty} . C_{\infty}$ is called the $e x-$ tended complex plane when the neighborhoods of $\infty$ (other than $C_{\infty}$ itself) are taken to be the complements in $C_{\infty}$ of the closed and bounded subsets (i.e., the compact subspaces) of $C$. These ideas add nothing to our understanding of the complex plane, but they do clarify many proofs and simplify the statements of many theorems, and they are valuable for this reason. Figure 27 gives an easy way of visualizing the extended complex plane. In this figure, the surface $S$ of a sphere of radius $1 / 2$ is rested tangentially on $C$ at the origin. It is customary to call the point of contact the south pole and the opposite point the north pole. The indicated projection from the north pole establishes a homeomorphism between $S$ minus its north pole and $C$, so from the topological point of view, $S$ minus its north pole can be regarded as essentially identical with the complex plane $C$. The north pole of $S$ can he considered to be the point at infinity, and passing from $C$ to $C_{\infty}$ amounts to using the point $\infty$ to plug up the hole in $C$ at the north pole. When $S$ is identified in this manner with the extended complex plane, it is usually called the Riemann sphere. In summary, the locally compact Hausdorff space $C$ has been made into the compact Hausdorff space $S$ by adding the single point $\infty$.

## 数学代写|拓扑学代写TOPOLOGY代 考|THE STONE-WEIERSTRASS THEOREMS

\left 的分隔符缺失或无法识别

(1) 的闭子代数 $\mathcal{C}[a, b]$ 由产生 $1, x$ 等于 $\mathcal{C}[a, b]$
(2) 的任何闭子代数 $\mathcal{C}[a, b]$ 其中包含 $1, x$ 等于 $\mathcal{C}[a, b]$.

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