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Box product of real lines does not have open connected components #1734

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Merged
felixpernegger merged 1 commit into from
Apr 10, 2026
Merged

Box product of real lines does not have open connected components #1734

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4 changes: 2 additions & 2 deletions spaces/S000107/properties/P000041.md → spaces/S000107/properties/P000234.md
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Original file line number Diff line number Diff line change
@@ -1,10 +1,10 @@
---
space: S000107
property: P000041
property: P000234
value: false
refs:
- mathse: 5012784
name: Answer to "Is $\ell^\infty$ with box topology connected?"
---

By {{mathse:5012784}} the connected component of an arbitrary point $x\in X$ is $A = \{y : y_n = x_n\text{ for all but finitely many }n\}$. Since $\text{int}(A) = \emptyset$, it follows that $x$ has no connected neighbourhoods.
By {{mathse:5012784}} the connected component of an arbitrary point $x\in X$ is $A = \{y : y_n = x_n\text{ for all but finitely many }n\}$. Since $\text{int}(A) = \emptyset$, it follows that $A$ is not open.
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