Pointed set

 ( Basic Concepts In Set Theory ) 

Maps between pointed sets $(X,\; x\_0)$ and $(Y,\; y\_0)$—called based maps,. pointed maps, or point-preserving maps—are functions from $X$ to $Y$ that map one basepoint to another, i.e. maps $f\; \backslash colon\; X\; \backslash to\; Y$ such that $f(x\_0)\; =\; y\_0$. Based maps are usually denoted $f\; \backslash colon\; (X,\; x\_0)\; \backslash to\; (Y,\; y\_0)$.

Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set $X$ together with a single nullary operation $*:\; X^0\; \backslash to\; X,$ which picks out the basepoint.

There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.

The category of pointed sets is a pointed category. The pointed $(\backslash \{a\backslash \},\; a)$ are both and , i.e. they are . The category of pointed sets and pointed maps has both products and , but it is not a distributive category. It is also an example of a category where $0\; \backslash times\; A$ is not isomorphic to $0$.