Identified
Undergraduate
Consider two objects that are not literally equal, but correspond to each other through a natural bijection. The objects are identified when we agree to treat the objects as the same.
Graduate (topology)
Consider some points of a topological space that we want to treat as one point. To identify those points is to pass to a quotient space in which they become a single point.
Examples
Examples
-
$\reals^2$ and $\mathbb{C}$, as real vector spaces
Undergraduate
$\reals^2$ and $\mathbb{C}$ are often identified through the linear bijection $(a, b) \mapsto a + b i$.
Distinctions involving this term
- Types of sameness: Mathematics has many ways to say that two objects are “the same.” Equal objects are the same object. The other words describe objects that are different but “the same in some respect,” and each one depends on what respect is meant.