The long ray is connected and locally connected, every proper initial segment is order-convex and connected, and, assuming countable choice, no at most countable subset is cofinal

Let $R = \omega_1 \times [0,1)$ be the closed long ray with its lexicographic order and its order topology (The closed long ray $\omega_1 \times [0,1)$ under the lexicographic order, and the long line, with the order topology, The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua), and let $0_R = (0,0)$ be its least element. For $x \in R$ write $[0_R, x] := \{\, y \in R : y \le x \,\}$ for the initial segment up to $x$. Then:

1. $R$ is connected, and its unique component is $R$ (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, Connected components, quasicomponents, and totally disconnected spaces).
2. Every initial segment $[0_R, x]$ is order-convex and connected, and so is every open ray and every interval of $R$.
3. $R$ is locally connected (Locally connected and locally path-connected spaces: a neighbourhood base of open connected, respectively open path-connected, sets at every point).
4. Assuming the Axiom of Countable Choice (The Axiom of Countable Choice ($\mathrm{AC}_\omega$)), no at most countable subset of $R$ is cofinal in $R$, that is, every at most countable subset has a strict upper bound (Finite, countably infinite, countable, uncountable).

Claim 4 is the order-theoretic analogue, transported to $R$, of the statement that no at most countable subset of $\omega_1$ is cofinal in $\omega_1$ (Assuming countable choice: every at most countable subset of $\omega_1$ is bounded below $\omega_1$, so no at most countable subset of $\omega_1$ is cofinal in it, and a supremum of at most countably many at most countable ordinals is at most countable, Cofinal subset of an ordinal); here a subset $D \subseteq R$ is called cofinal when for every $x \in R$ there is $y \in D$ with $x \le y$.

Path-connectedness is not asserted. Whether $R$ is path-connected is not settled by any item among this page's declared prerequisites, and nothing here claims it either way. Consequently the path components of $R$ are not computed.

• Local connectedness is immediate here and is not a deep property of $R$. Every basic open set of an order topology is order-convex, and in a linear continuum every order-convex set is connected. So any linear continuum is locally connected, and $R$ inherits that with no reference to $\omega_1$.

• What distinguishes $R$ from an ordinary half-line is claim 4 alone. The first three claims hold verbatim for $[0,\infty) \subseteq \mathbb{R}$, which is also a linear continuum with a least element and no greatest. In $[0,\infty)$ the at most countable set of canonical naturals is cofinal; in $R$ no at most countable set is, and that is the whole content of the word long.

• The choice cost is inherited and is not spent again here. Claim 4 uses claim 3 of The long ray is a linear continuum, hence connected; every one of its at most countable subsets is bounded above, assuming countable choice, whose own statement carries $\mathrm{AC}_\omega$; the argument above adds only the passage from an upper bound to a strict one, which needs nothing beyond $R$ having no greatest element.