Any two points in a connected smooth manifold can be joined by a piecewise c one curve

Given: A connected nonempty smooth manifold and points $p,q$.

Piecewise c one curve on a manifold: A piecewise $C^1$ curve in $M$ is a continuous map $\gamma :[a,b]\to M$ with a finite subdivision such that its restriction to each closed piece is $C^1$ in local charts, with one-sided derivatives at piece endpoints. Use the chartwise regularity convention of def-c-r-and-smooth-maps-between-smooth-manifolds and the finite path operations of def-piecewise-c1-path-operations-and-oriented-reparametrizations. Refining a piece into finitely many chart pieces is allowed. No nonzero-velocity hypothesis is imposed: constant segments and pauses are admissible. A singleton parameter interval is interpreted as a constant curve of length zero.

Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets: Let $(X,\mathcal{T})$ be a topological space (def-topological-space). - A separation of $X$ is an ordered pair $(U,V)$ of open, nonempty, disjoint subsets of $X$ with $U\cup V=X$. - $X$ is disconnected when a separation of $X$ exists, and connected when none does. - A subset $A\subseteq X$ is a connected subset of $X$ when the space $(A,{\mathcal{T}}_A)$ is connected, ${\mathcal{T}}_A$ being the subspace topology (def-subspace-topology-top). "Disconnected subset" is read the same way. Since $U$ and $V$ are complementary in $X$, each of them is closed as well as open; so a separation is the same thing as a partition of $X$ into two nonempty clopen pieces (def-topological-space). The clopen subsets of $X$ are those that are both open and closed, and $\mathrm{\varnothing }$ and $X$ are always among them. The empty space and the one-point space are connected in this library. Neither admits a separation: a separation requires two nonempty disjoint sets whose union is the whole space, and neither $\mathrm{\varnothing }$ nor a singleton can be written as such a union. So both are connected under the definition above, without any special clause. This is a live convention fork and the competing choice is recorded in rem-connectedness-conventions; nothing on this page depends on which is taken except the reading of the word "connected" applied to those two spaces. Connectedness is a property of a space, not of an ambient pair. The condition above mentions only $(X,\mathcal{T})$. When it is applied to $A\subseteq X$ it is applied to the space $(A,{\mathcal{T}}_A)$, so it does not change if $A$ is regarded as a subspace of some other space inducing the same topology on $A$; in particular a subset of $A$ is connected as a subset of $A$ exactly when it is connected as a subset of $X$, by transitivity of the subspace topology (def-subspace-topology-top). This is why "connected" may be used of a subset with no ambient space named. Spelled out for a subset. $A\subseteq X$ is disconnected exactly when there are open $U,V\subseteq X$ with $A\subseteq U\cup V,U\cap A\mathrm{\ne }\mathrm{\varnothing },V\cap A\mathrm{\ne }\mathrm{\varnothing },U\cap V\cap A=\mathrm{\varnothing },$ because the open sets of $(A,{\mathcal{T}}_A)$ are precisely the traces $U\cap A$. Note the last condition: it asks $U$ and $V$ to be disjoint on $A$, not in $X$. Requiring $U\cap V=\mathrm{\varnothing }$ outright is a strictly stronger demand and is a different notion. The two-point discrete space. Write $\mathbf{2}:=\{0,1\}$ with the discrete topology (def-standard-topologies), in which every subset is open. A separation of $X$ is the same datum as a surjective continuous map $X\to \mathbf{2}$ (def-continuous-map-top): given $(U,V)$, the map sending $U$ to $0$ and $V$ to $1$ is continuous because the preimage of each of the four open subsets of $\mathbf{2}$ is one of $\mathrm{\varnothing }$, $U$, $V$, $X$; given a surjective continuous $\chi :X\to \mathbf{2}$, the pair $({\chi }^{-1}[\{0\}],{\chi }^{-1}[\{1\}])$ is a separation. This reformulation is proved as a theorem on this page and is recorded here only to name $\mathbf{2}$. Separated sets. Two subsets $A_1,A_2\subseteq X$ are separated in $X$ when $\overline{A_1}\cap A_2=\mathrm{\varnothing }\text{and}{A}_1\cap \overline{A_2}=\mathrm{\varnothing },$ closures taken in $X$ (def-interior-closure-boundary-top, thm-closure-characterisation-top). Separated sets are disjoint, since $A_1\subseteq \overline{A_1}$; the converse fails. This is verbatim the condition def-connected-r uses on the real line, transported to an arbitrary space, and the theorem relating it to the definition above is the next lemma on this page. Totally disconnected spaces, and the empty case. The vocabulary for a space all of whose connected subsets are single points is fixed later on this page, together with the components; it is not defined here because it is stated in terms of components.