拓扑空间的邻域刻画：定义及邻域系的性质定理

Hey everyone, let's dive into a core foundational concept in topology: neighborhoods and their associated neighborhood systems. Here's a clear, structured breakdown of the definition and key properties:

Definition #

Let $(X, \mathcal{T})$ be a topological space. For a point $x \in X$, a subset $V \subseteq X$ is called a neighborhood of $x$ if there exists an open set $G \in \mathcal{T}$ such that $x \in G \subseteq V$. We denote $\mathcal{V}(x)$ as the collection of all neighborhoods of $x$.

Key Theorem: Properties of the Neighborhood System $\mathcal{V}(x)$ #

Given a topological space $X$ and a point $x \in X$, the neighborhood system $\mathcal{V}(x)$ satisfies the following three axioms:
$$(V_1): X \in \mathcal{V}(x)$$
$$(V_2): \forall V \in \mathcal{V}(x): x \in V$$
$$(V_3): \forall V,W \in \mathcal{V}(x): V \cap W \in \mathcal{V}(x)$$

内容的提问来源于stack exchange，提问作者user370967