---- > [!definition] Definition. ([[contractible]]) > A [[topological space]] $X$ is said to be **contractible** if the [[identity map]] $\id_{X}$ is [[homotopy|nulhomotopic]]. Put differently, the [[contractible]] spaces are exactly those which [[deformation retract]] to a point. > [!basicexample] >$\mathbb{R}^{n}$ is contractible to any constant $\b c$. This is witnessed by the [[straight-line homotopy]] $\begin{align} F: \mathbb{R}^{n} \times I \to \b c \\ t \b c + (1-t)\b x \end{align}.$ > [!basicproperties] > Contractible spaces are [[simply connected]]. For [[deformation retract|deformation retraction]] preserves [[homotopy equivalent|homotopy type]] and thus [[fundamental group]] [[group isomorphism|isomorphism class]], and any singleton has trivial [[fundamental group]]. ^b9bb0b ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch04