Is ℝ uniquely determined by the “point removal splits it into two copies of itself” property?

Question

Let $X$ be a non-empty topological space satisfying the following conditions. Is $X$ homeomorphic to the topological space $\mathbb{R}$?

• $X$ is $T_1$.
• $X$ is connected.
• For any point $x$ in $X$, the space $X \setminus \{x\}$ is homeomorphic to the disjoint union $X \sqcup X$ (with the disjoint union topology).

In other words, I wonder whether the topological space $\mathbb R$ can be characterized by the property that “removing a single point results in two copies of the space.” If this is true, I think it would be an interesting characterization of ℝ, so I am curious about it.

The $T_1$ condition was included to exclude some pathological examples, so I would also welcome answers that remove the $T_1$ condition or replace it with other separation axioms. I also welcome answers about similar characterizations of ℝ.

Thanks in advance!

Answer 1

Closely related issues are discussed in this mathoverflow post, where the following statement is referenced, with stronger separation hypotheses and local hypotheses but with a weaker "removing a single point" property:

The real line is characterized up to homeomorphism as a connected, locally connected, separable, regular space $X$ such that for any $x \in X$ the subspace $X-\{x\}$ has exactly two connected components.

Answer 2

Here is a counter-example:

Let $(X, \le)$ be an infnite, linearly ordered set, $\tau = $ order topology on $X$. Hence, $X$ is $T_1$ (in fact, hereditarily normal, $T_1$).

In this paper from Akin, Hrbacek $X$ is called CHLOTS, if it is complete (i.e., every bounded, non-empty subset has a supremum) and order isomorphic with every nonempty, open, convex subset of itself (p. 42).

Let $(X, \le)$ be CHLOTS. Obviously, $X$ has no minimum and no maximum and is dense-in-itself (i.e., $(x, y) \neq \emptyset$ for all $x < y$), hence, by completeness, $(X, \tau)$ is connected.
For $x \in X$, $X^{<x} = \{y \in X: y < x\}$ is order isomorphic to $X$. Since $X^{<x}$ is convex, its order topology coincides with the subspace topology, hence $X^{<x}$ is homeomorphic to $X$ and $X^{<x}$ is clopen in $X\setminus\{x\}$. Analogously for $X^{>x}$. Hence, $X\setminus \{x\}$ is the topological sum of two copies of $X$. Thus, all three of the above conditions hold for $(X, \tau)$.

The above referenced paper gives a lot of examples of CHLOTS, which are not order isomorphic, hence not homeomorphic, to $\mathbb R$, for instance $X = \mathbb R \times [0,1] \times [0,1] \times \ldots$ with the lexicographic ordering (Theorem 4.2, see also remark (1.4) on p. 2).

Remark
A connected LOTS is also locally connected. Hence, the above example lacks exactly separability from the characterzing properties given in the answer of Lee Mosher.