拓扑空间是有序对 [latex]（X，\tau）[/latex]，其中 [latex]X[/latex] 是一个集合，[latex]\tau[/latex] 是 [latex]X[/latex] 的子集集合，称为开集，满足三个公理：1）空集 [latex]\emptyset[/latex] 和 [latex]X[/latex] 本身都在 [latex]\tau[/latex] 中。2) [latex]\tau[/latex] 中任意多个集合的联合也在 [latex]\tau[/latex] 中。3）[latex]\tau[/latex] 中任意有限个集合的交集也在 [latex]\tau[/latex] 中。
The collection [latex]\tau[/latex] is called a topology on [latex]X[/latex]. The elements of [latex]X[/latex] are usually called points, and the subsets in [latex]\tau[/latex] are the open sets. A subset of [latex]X[/latex] is called closed if its complement is an open set. This axiomatic definition is extremely general and powerful, allowing for the study of spatial properties in a way that is independent of distance or measurement. For example, the set of real numbers [latex]\mathbb{R}[/latex] with the collection of all open intervals forms a topological space, known as the standard topology. However, many other, non-standard topologies can be defined on the same set [latex]\mathbb{R}[/latex]. The concept of a neighborhood of a point is fundamental; a neighborhood of a point [latex]x[/latex] is any subset of [latex]X[/latex] that contains an open set which in turn contains [latex]x[/latex]. This framework allows mathematicians to generalize concepts like limits and continuity from metric spaces to more abstract settings. The power of this definition lies in its ability to capture the essence of ‘closeness’ and ‘connectedness’ without relying on a metric, which makes it applicable to a vast range of mathematical and scientific problems where a notion of distance is not natural or available.
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