Topology

If $p$ is a path in $X$ from $x_0$ to $x_1\text{,}$ then ${\beta }_p:{\pi }_1(X,x_0)\to {\pi }_1(X,x1)$ is an isomorphism of groups.

Let $X$ be a topological space and let $x_0,x_1\in X\text{.}$ Let $h:X\to Y$ be a continuous function, and let $h_0:(X,x_0)\to (Y,h(x_0))$ and $h_1:(X,x1)\to (Y,h(x1))$ be the corresponding maps of pointed spaces. If $p:I\to X$ is a path from $x_0$ to $x_1\text{,}$ then $\beta h\circ p\circ (h_0)\ast =(h_1)\ast \circ \beta p\text{.}$

If $X$ and $Y$ are path-connected spaces and $X\simeq Y\text{,}$ then ${\pi }_1(X)\cong {\pi }_1(Y)\text{.}$

If $X$ and $Y$ are PC topological spaces and $X\times Y$ has the product topology, then ${\pi }_1(X\times Y)\cong {\pi }_1(X)\times {\pi }_1(Y)\text{.}$

If $r:X\to A$ is a retraction and $i:A\to X$ is the inclusion, then $r\ast$ is onto and $i\ast$ is one-to-one.

Let $A$ be a PC subspace of a PC topological space $X\text{,}$ and let $a_0\in A\text{.}$ Let $i:A\to X$ be the inclusion map, and let $i\ast :{\pi }_1(A,a_0)\to {\pi }_1(A,a_0)$ be the induced homomorphism. Then $i\ast$ is onto if and only if every path in $X$ with endpoints in $A$ is path homotopic to a path in $A\text{.}$