---- > [!definition] Definition. ([[topological manifold]]) > A **topological $k$-manifold** is a [[second-countable space|second-countable]] [[Hausdorff space]] $M$ such that each point $p \in M$ has a [[neighborhood]] that is [[homeomorphism]] with an open subset of $\mathbb{R}^{k}$. > [!note] Remark. > Second-countability is needed for Euclidean embeddings. > [!basicnonexample] > The 'cross [[subspace topology|subspace]]' $X=\{ (x,y) \in \mathbb{R}^{2} : xy=0 \}$ of $\mathbb{R}^{2}$ is not a $k$-manifold for any $k \in \mathbb{N}$. > First note that $X$ is [[connected]] (for it is obviously [[path-connected]]). > > Consider $\begin{align} X - \{ \b 0 \}= & (-\infty, 0) \times \{ 0 \} \\ & \sqcup (0, \infty) \times \{ 0 \} \\ & \sqcup \{ 0 \} \times (-\infty , 0) \\ & \sqcup \{ 0 \} \times (0,\infty). \end{align}$> So $X - \{ \b 0 \}$ is a disjoint union of open sets. Moreover, each constituent set is [[connected]]: [[linear continuums in the order topology and the intervals they contain are connected|the rays]] $(-\infty, 0)$ and $(0, \infty)$ in $\mathbb{R}$ are [[connected]] and singletons are too; then recall that [[a finite cartesian product of connected spaces is connected]] (I guess we implicitly use here [[product and subspace topologies commute]]). These sets are exactly the [[connected component]]s of $X$, since $\{ 0 \}$ is the only outside [[limit point]] of each (and so it is easy to construct a [[separation of a topological space|separation]] of any prospective [[connected component]] that takes on values in more than one of these sets). This means $X-\{ \b 0 \}$ is not [[connected]]; it is the disjoint union of four [[connected component|connected components]]. > So we cannot have $k\geq 2$, that would contradict the result that [[2-manifolds and up remain connected when you remove a point]]. We cannot have $k=1$ either though, because [[2-manifolds and up remain connected when you remove a point|that would contradict]] the result that [[connected]] 1-manifolds have at most two connected components. ^2967c6 ---- #### ---- #### 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 > ```