Britannica Sources used in this answer

How does a subset inherit a relative topology from a larger topological space?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In a topological space, a subset $A$ of a space $X$ inherits a structure known as the relative topology (or subspace topology) from $X$ .

To define this topology, the open sets of $A$ are determined by taking the intersection of $A$ with each of the open sets defined in the original space $X$ . Formally, if $T$ is the collection of open sets in $X$ , the relative topology on $A$ consists of all sets of the form $A \cap U$ , where $U \in T$ .1

This construction allows mathematicians to treat subsets as topological spaces in their own right, facilitating the study of how properties of the larger space $X$ restrict to the subset $A$ .1