双连续的

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 定义

再给出一个定义（这不会是最后一个）。拓扑空间 $(M,\tau )$ 与 $(\tilde{M},\tilde{\tau })$ 之间的一个同胚(homeomorphism)，是指一个映射 $f:M\to \tilde{M}$，它满足：

i) 单射的(injective)（或称一一的）：即对 $p\ne q,f(p)\ne f(q)$

ii) 满射的(surjective)（或称映上的）：$f(M)=\tilde{M}$，也就是说，对每个 $\tilde{p}\in \tilde{M}$，都存在一个 $p\in M$，使得 $f(\mathbf{p})=\tilde{\mathbf{p}}.$

既单射又满射的函数称为双射的(bijective)。这保证了它们有逆映射

iii) 双连续的(bicontinuous)。这意味着该函数及其逆映射都是连续的。要定义连续性的概念，我们需要用到拓扑。我们说 $f$ 是连续的，如果对所有 $\tilde{\mathcal{O}}\in \tilde{\mathcal{T}},f^{-1}(\tilde{\mathcal{O}})\in \mathcal{T}$

有一个甜甜圈变成咖啡杯、又变回甜甜圈的动画，常被用来图解拓扑学的思想。如果你想说得花哨一点，可以说甜甜圈与咖啡杯同胚。

英文原文

One further definition (it won’t be our last). A homeomorphism between topological spaces $(M,\tau )$ and $(\tilde{M},\tilde{\tau })$ is a map $f:M\to \tilde{M}$ which is

i) Injective (or one-to-one): for $p\ne q,f(p)\ne f(q)$

ii) Surjective (or onto): $f(M)=\tilde{M}$ , which means that for each $\tilde{p}\in \tilde{M}$ there exists a $p\in M$ such that $f(\mathbf{p})=\tilde{\mathbf{p}}.$

Functions which are both injective and surjective are said to be bijective. This ensures that they have an inverse

iii) Bicontinuous. This means that both the function and its inverse are continuous. To define a notion of continuity, we need to use the topology. We say that $f$ is continuous if, for all $\tilde{\mathcal{O}}\in \tilde{\mathcal{T}},f^{-1}(\tilde{\mathcal{O}})\in \mathcal{T}$

There’s an animation of a donut morphing into a coffee mug and back that is often used to illustrate the idea of topology. If you want to be fancy, you can say that a donut is homeomorphic to a coffee mug.

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