A doubt regarding the proof of Theorem 24.1 in Munkres' "Topology".

Question

On pg. 249 of Munkres' "Topology", Theorem 24.1 says that if $L$ is a linear continuum in the order topology, then $L$ is connected.

He then proves this for every convex subspace of $L$.

I don't understand how the whole of $L$ can be connected, if we have only proved that convex subsets of $L$ are connected. Can we prove that every linear continuum ($L$ in this case) can be expressed as the union of non-disjoint convex subsets?

Thank you.

$L$ is a convex subset/subspace of itself. – user642796 Mar 17 '14 at 11:38 — user642796, Mar 17 '14 at 11:38
Does this answer your question? Linear continuum is convex – tryst with freedom Sep 05 '22 at 13:19 — tryst with freedom, Sep 05 '22 at 13:19

score 1 · Answer 1 · answered Mar 17 '14 at 17:29

1

$L$ is a convex subset of itself, trivially. Munkres thus proves a more general fact.

answered Mar 17 '14 at 17:29

Henno Brandsma

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A doubt regarding the proof of Theorem 24.1 in Munkres' "Topology".

1 Answers1

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