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I am trying to solve

What are the path components of $\mathbb{R}_l$?

We know that $\mathbb{R}_l$ is lower limit topology.

Definition of path component here

I am thinking that the path components of $\mathbb{R}_l$ are singleton sets.

MY ATTEMPT:

Suppose it is not, let $A$ be a path connected subspace of $\mathbb{R}_l$ with more than one point. Let $a, b \in A$ such that $a<b$. Then the sets $A \cap (-\infty, b)$ and $A \cap [b, +\infty)$ are open in $A$, and clearly $a \in A \cap (-\infty, b)$ and $b \in A \cap [b, +\infty)$, hence nonempty, and also $A = (A \cap (-\infty, b)) \sqcup (A \cap [b, +\infty))$

But I have no idea how to proceed further. Please help me.

José Carlos Santos
  • 427,504
  • See here https://math.stackexchange.com/questions/1770696/connected-components-for-bbb-r-mathcal-t-lower-limit and here https://math.stackexchange.com/questions/172964/is-r-with-j-d-topology-totally-disconnected – Sumanta Aug 05 '21 at 12:08

1 Answers1

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You are right: the only connected subsets of $\Bbb R_l$ are the singletons; in other words, $\Bbb R_l$ is totally disconnected.

Take $a,b\in\Bbb R$, with $a<b$, and suppose that there is a connected subset $C$ of $\Bbb R_l$ such that $a,b\in C$. Let $c\in(a,b)$. Then both $[c,\infty)$ and $[a,c)$ are clopen sets. In particular, $[c,\infty)\cap C$ and $[a,c)\cap C$ are non-empty disjoint subsets of $C$ whose union is $C$. This is impossible, since $C$ is connected.

José Carlos Santos
  • 427,504
  • Sir, thanks for your answer. But I am asking about path components of $\mathbb{R}_l$. –  Aug 05 '21 at 11:19
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    Since $\Bbb R_l$ has no connected subsets with more than one element, then, in particular, it has no path-connected subsets with more than one element. – José Carlos Santos Aug 05 '21 at 11:22