如果你也在 怎样代写拓扑学topology这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。拓扑学topology在数学中，拓扑学（来自希腊语中的τόπος，”地方、位置”，和λόγος，”研究”）关注的是几何对象在连续变形下保持的属性，如拉伸、扭曲、皱缩和弯曲；也就是说，在不关闭孔、打开孔、撕裂、粘连或穿过自身的情况下。
拓扑学topology拓扑空间是一个被赋予了结构的集合，称为拓扑，它允许定义子空间的连续变形，以及更广泛地定义所有种类的连续性。欧几里得空间，以及更一般的，公制空间都是拓扑空间的例子，因为任何距离或公制都定义了一个拓扑。拓扑学中所考虑的变形是同构和同形。在这种变形下不变的属性是一种拓扑属性。拓扑属性的基本例子有：维度，它可以区分线和面；紧凑性，它可以区分线和圆；连接性，它可以区分一个圆和两个不相交的圆。
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我们提供的拓扑学topology及其相关学科的代写，服务范围广, 其中包括但不限于:
数学代写|拓扑学作业代写topology代考|Topologies on spaces of smooth maps
Let $M, N$ be smooth manifolds. We define the weak topology on every set $\mathcal{C}^{r}(M, N), r \geq 0$, the topological spaces $\mathcal{E}^{r}(M, N)$ (the subspaces of $\mathcal{C}^{r}(M, N)$ formed by the smooth maps) and the space $\mathcal{E}(M, N)$ (that is, $\mathcal{C}^{\infty}(M, N)$ equipped with the union of the $\mathcal{E}^{r}$-topologies $)$. This extends the special case of open sets treated in Chapter 1. As in that case, we provide a basis of open neighbourhoods of every element in the pertinent map space. For every $f \in \mathcal{C}^{r}(M, N)$, we consider neighbourhoods of the form
$$
\mathcal{U}{r}\left(f, f{U, U^{\prime}}, K, \epsilon\right)
$$
where:
- $f_{U, U^{\prime}}: U^{\prime} \rightarrow U$ is a (necessarily $\mathcal{C}^{r}$ ) representation of $f$ in local coordinates $\left(U \subset \mathbb{R}^{m}, U^{\prime} \subset \mathbb{R}^{n}\right.$ being open sets).
- $K \subset U$ is a compact set.
- $\epsilon>0$.
Then $g \in \mathcal{C}^{r}(M, N)$ belongs to
$$
\mathcal{U}{r}\left(f, f{U, U^{\prime}}, K, \epsilon\right)
$$
if and only if it admits a local representation (over the same open sets $U, U^{\prime}$ ) $g_{U, U^{\prime}}: U \rightarrow U^{\prime}$ such that $g_{U, U^{\prime}} \in \mathcal{U}{r}\left(f{U, U^{\prime}}, K, \epsilon\right) \subset \mathcal{C}^{k}\left(U, U^{\prime}\right)$
If $M \subset \mathbb{R}^{h}, N \subset \mathbb{R}^{k}$ are embedded manifolds, there is an equivalent way to define these topologies. For every $f \in \mathcal{C}^{r}(M, N)$ we consider neighbourhoods of the form
$$
\mathcal{U}_{r}(f, \hat{f}, K, \epsilon)
$$
where:
- $\hat{f}: \Omega \rightarrow \mathbb{R}^{k}$ is a local $\mathcal{C}^{r}$ extension of $f_{\mid W}: W \rightarrow N, W=\Omega \cap M$, $\Omega \subset \mathbb{R}^{h}$ being open.
- $K \subset W$ is a compact set.
- $\epsilon>0$.
Then $g \in \mathcal{C}^{r}(M, N)$ belongs to $\mathcal{U}{r}(f, \hat{f}, K, \epsilon)$ if and only if there exists a $\mathcal{C}^{r}$ extension $\hat{g}: \Omega \rightarrow \mathbb{R}^{k}$ of $g{\mid W}$ such that $\hat{g} \in \mathcal{U}_{r}(\hat{f}, K, \epsilon) \subset \mathcal{C}^{r}\left(\Omega, \mathbb{R}^{k}\right)$.
数学代写|拓扑学作业代写topology代考|Homotopy, isotopy, diffeotopy, homogeneity
These notions, already introduced in Chapter 1 in the special case of open sets, extend verbatim to smooth manifolds. They correspond to continuous paths in suitable map spaces and carry equivalence relations.
The proof of the homogeneity theorem, Theorem 1.18, is essentially local and extends straightforwardly.
Theorem 4.9. Let $N$ be a connected smooth manifold. Let $p, q \in N$. Then there is a diffeotopy with compact support between $f_{0}=\mathrm{id}{N}$ and $f=f{1}$ such that $f(p)=q$.
数学代写|拓扑学作业代写TOPOLOGY代考|The (abstract) tangent functor
For embedded manifolds, tangent bundles and maps have been constructed as a direct generalization of the basic case of open sets in Euclidean spaces. For abstract manifolds, they must be somehow “invented”, with the constraint that they must be compatible with what is already done in the embedded category. This will lead us in Section $4.4$ to the general notion of fibre bundle in the sense of Steenrod [Steen].
4.3.1. Fibre bundles. The tangent vector bundle of an embedded manifold is the first fundamental example of the general notion of fibre bundle. We saw other examples in Chapter 3, dealing with Grassmann and Stiefel manifolds. As we will find many more examples, at this point it is convenient to formalize this notion.
拓扑学代写
数学代写|拓扑学作业代写TOPOLOGY代考|TOPOLOGIES ON SPACES OF SMOOTH MAPS
Let $M, N$ be smooth manifolds. We define the weak topology on every set $\mathcal{C}^{r}(M, N), r \geq 0$, the topological spaces $\mathcal{E}^{r}(M, N)$ (the subspaces of $\mathcal{C}^{r}(M, N)$ formed by the smooth maps) and the space $\mathcal{E}(M, N)$ (that is, $\mathcal{C}^{\infty}(M, N)$ equipped with the union of the $\mathcal{E}^{r}$-topologies $)$. This extends the special case of open sets treated in Chapter 1. As in that case, we provide a basis of open neighbourhoods of every element in the pertinent map space. For every $f \in \mathcal{C}^{r}(M, N)$, we consider neighbourhoods of the form
$$
\mathcal{U}{r}\left(f, f{U, U^{\prime}}, K, \epsilon\right)
$$
其中：
- F在,在′:在′→在是一个n和C和ss一种r一世l是$Cr$的代表F在当地坐标(在⊂R米,在′⊂Rn是开集）。
- ķ⊂在是紧集。
- ε>0.
然后G∈Cr(米,ñ)属于
$$
\mathcal{U} {r}\left(f, f {U, U^{\prime}}, K, \epsilon\right)
$$
当且仅当它承认局部表示这在和r吨H和s一种米和这p和ns和吨s$在,在′$ G在,在′:在→在′这样 $g_{U, U^{\prime}} \in \mathcal{U} {r}\left(f {U, U^{\prime}}, K, \epsilon\right) \subset \mathcal {C}^{k}\左U, U^{\素数}\右U, U^{\素数}\右$
如果米⊂RH,ñ⊂Rķ是嵌入式流形，有一种等效的方法来定义这些拓扑。对于每一个F∈Cr(米,ñ)我们考虑以下形式的邻域
在r(F,F^,ķ,ε)
在哪里：
- $\hat{f}: \Omega \rightarrow \mathbb{R}^{k}$ is a local $\mathcal{C}^{r}$ extension of $f_{\mid W}: W \rightarrow N, W=\Omega \cap M$, $\Omega \subset \mathbb{R}^{h}$ being open.
- $K \subset W$ is a compact set.
- $\epsilon>0$.
- Then $g \in \mathcal{C}^{r}(M, N)$ belongs to $\mathcal{U}{r}(f, \hat{f}, K, \epsilon)$ if and only if there exists a $\mathcal{C}^{r}$ extension $\hat{g}: \Omega \rightarrow \mathbb{R}^{k}$ of $g{\mid W}$ such that $\hat{g} \in \mathcal{U}_{r}(\hat{f}, K, \epsilon) \subset \mathcal{C}^{r}\left(\Omega, \mathbb{R}^{k}\right)$.
数学代写|拓扑学作业代写TOPOLOGY代考|HOMOTOPY, ISOTOPY, DIFFEOTOPY, HOMOGENEITY
这些概念已经在第 1 章的开集特例中介绍过，逐字扩展至光滑流形。它们对应于合适的地图空间中的连续路径，并带有等价关系。
同质性定理的证明，定理 1.18，本质上是局部的并且直接扩展。
定理 4.9。让ñ是一个连通的光滑流形。让p,q∈ñ. 然后在 $f_{0}=\mathrm{id} {N}之间有一个紧凑支持的微分拓扑一种ndf=f {1}s在CH吨H一种吨Fp=q$。
数学代写|拓扑学作业代写TOPOLOGY代考|THE 一种bs吨r一种C吨正切函子
对于嵌入式流形，切线束和映射已被构建为欧几里得空间中开集基本情况的直接推广。对于抽象流形，它们必须以某种方式“发明”，但约束条件是它们必须与嵌入类别中已经完成的内容兼容。这将引导我们进入部分4.4Steenrod 意义上的纤维束的一般概念小号吨和和n.
4.3.1。纤维束。嵌入流形的切向量丛是纤维丛一般概念的第一个基本例子。我们在第 3 章看到了其他例子，处理 Grassmann 和 Stiefel 流形。正如我们会发现更多的例子，在这一点上，将这个概念形式化是很方便的。
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