| Version 7 |
Version 6 |
| A \emph{pointed topological space}, written as $(X,x_0)$, consists |
A \emph{pointed topological space}, written as $(X,x_0)$, consists |
| of a non-empty topological space $X$ together with an element |
of a non-empty topological space $X$ together with an element |
| $x_0\in X$. The terminology \emph{based topological space} is also |
$x_0\in X$. The terminology \emph{based topological space} is also |
| used often. |
used often. |
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| If $(X,x_0)$ is a pointed space, we call $X$ its \emph{underlying} |
If $(X,x_0)$ is a pointed space, we call $X$ its \emph{underlying} |
| topological space and $x_0$ its \emph{basepoint}. |
topological space and $x_0$ its \emph{basepoint}. |
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| A \emph{morphism} from $(X,x_0)$ to $(Y,y_0)$ is a continuous map |
A \emph{morphism} from $(X,x_0)$ to $(Y,y_0)$ is a continuous map |
| $f\co X\to Y$ satisfying $f(x_0)=y_0$. With these morphisms, the |
$f\co X\to Y$ satisfying $f(x_0)=y_0$. With these morphisms, the |
| pointed topological spaces form a category. |
pointed topological spaces form a category. |
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| Two pointed topological spaces $(X,x_0)$ and $(Y,y_0)$ are |
Two pointed topological spaces $(X,x_0)$ and $(Y,y_0)$ are |
| isomorphic in this category if there exists a homeomorphism |
isomorphic in this category if there exists a homeomorphism |
| $f\co X\to Y$ with $f(x_0)=y_0$. |
$f\co X\to Y$ with $f(x_0)=y_0$. |
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| Every singleton (a pointed topological space of the form $(\{x_0\}, |
Every singleton (a pointed topological space of the form $(\{x_0\}, |
| x_0)$) is a zero object in this category. |
x_0)$) is a zero object in this category. |
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| For every pointed topological space $(X,x_0)$, we can construct the |
For every pointed topological space $(X,x_0)$, we can construct the |
| fundamental group $\pi(X,x_0)$ and for every morphism |
fundamental group $\pi(X,x_0)$ and for every morphism |
| $f\co (X,x_0)\to(Y,y_0)$ we obtain a group homomorphism |
$f\co (X,x_0)\to(Y,y_0)$ we obtain a group homomorphism |
| $\pi(f)\co\pi(X,x_0)\to \pi(Y,y_0)$. This yields a functor from the |
$\pi(f)\co\pi(X,x_0)\to \pi(Y,y_0)$. This yields a functor from the |
| category of pointed topological spaces to the category of groups. |
category of pointed topological spaces to the category of groups. |
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| Other interesting functors defined on the category of pointed spaces |
Other interesting functors defined on the category of pointed spaces |
| include the higher homotopy groups $\pi_i(X,x_0)$ for $i=2,3,\ldots$ |
include the higher homotopy groups $\pi_i(X,x_0)$ for $i=2,3,\ldots$ |
| that map into the category of abelian groups and the (based) |
that map into the category of abelian groups and the (based) |
| \emph{loop space} $\Omega(X,x_0)$ that maps into the category of |
\emph{loop space} $\Omega(X,x_0)$ that maps into the category of |
| topological spaces. |
topological spaces. |
