identity mapIdentityMap2003-11-01 16:21:142006-10-15 17:09:21Definition% this is the default PlanetMath preamble. as your knowledge
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\newcommand*{\norm}[1]{\lVert #1 \rVert}
\newcommand*{\abs}[1]{| #1 |}{\bf Definition}
If $X$ is a set, then the {\bf identity map} in $X$ is the mapping
that maps each element in $X$ to itself.
\subsubsection{Properties}
\begin{enumerate}
\item An identity map is always a bijection.
\item Suppose $X$ has two topologies $\tau_1$ and $\tau_2$. Then
the identity mapping $I:(X,\tau_1)\to (X,\tau_2)$ is continuous if and only if
$\tau_1$ is finer than $\tau_2$, i.e., $\tau_1\subset\tau_2$.
\item
The identity map on the $n$-sphere, is
\PMlinkname{homotopic}{HomotopyOfMaps}
to the antipodal map $A:S^n\to S^n$ if $n$ is odd \cite{guillemin}.
\end{enumerate}
\begin{thebibliography}{9}
\bibitem{guillemin} V. Guillemin, A. Pollack,
\emph{Differential topology}, Prentice-Hall Inc., 1974.
\end{thebibliography}