----- > [!proposition] Proposition. ([[a finite cartesian product of connected spaces is connected]]) > Finite [[cartesian product|cartesian products]] of [[connected]] [[topological space|topological spaces]] is [[connected]]. > \ > This result extends to infinite products so long as the product set is endowed with the [[product topology]]. (It does not, for example, hold for the [[box topology]]). > [!proof]- Proof. ([[a finite cartesian product of connected spaces is connected]]) > We prove the theorem first for the product of two [[connected]] spaces, $X$ and $Y$. Choose a "base point" $(a,b) \in X \times Y$. The 'slice' $X \times \{ b \}$, being [[homeomorphism|homeomorphic]] to $X$, is [[connected]] ([[homeomorphisms preserve structure]]). Likewise $Y \times \{ x \}$ is [[connected]] for all $x \in X$ since it is [[homeomorphisms preserve structure|homeomorphic]] to $Y$. This yields a 't-shaped space' $t_{x}:=(X \times \{ b \}) \cup (\{ x \} \times Y) .$ Each $t_{x}$ contains the point $(a,b)$, and the union $\bigcup_{x \in X}^{}t_{x}$ is [[connected]] by [[arbitrary union of nontrivially-intersecting connected subspaces is connected]] ($(a,b)$ witness the nontriviality of the intersection). This union equals $X \times Y$. > In the general case we can proceed via induction with the same general approach. ----- #### ---- #### 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 > ```