Skip to main content
MathOverflow

Return to Question

replaced http://mathoverflow.net/ with https://mathoverflow.net/
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the Lebesgue covering dimension equal 1 as showed in Topological spaces with Lebesgue covering dimension 1Topological spaces with Lebesgue covering dimension 1. However, the topological space presented in that post does not have a basis of connected sets. My question is: If $X$ is a connected topological space with Lebesgue covering dimension 1 having a basis of connected subsets then all connected subset (not a singleton) has no empty interior?

We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the Lebesgue covering dimension equal 1 as showed in Topological spaces with Lebesgue covering dimension 1. However, the topological space presented in that post does not have a basis of connected sets. My question is: If $X$ is a connected topological space with Lebesgue covering dimension 1 having a basis of connected subsets then all connected subset (not a singleton) has no empty interior?

We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the Lebesgue covering dimension equal 1 as showed in Topological spaces with Lebesgue covering dimension 1. However, the topological space presented in that post does not have a basis of connected sets. My question is: If $X$ is a connected topological space with Lebesgue covering dimension 1 having a basis of connected subsets then all connected subset (not a singleton) has no empty interior?

Source Link
Didi
Didi
  • 95
  • 4

One-dimensional topological spaces

We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the Lebesgue covering dimension equal 1 as showed in Topological spaces with Lebesgue covering dimension 1. However, the topological space presented in that post does not have a basis of connected sets. My question is: If $X$ is a connected topological space with Lebesgue covering dimension 1 having a basis of connected subsets then all connected subset (not a singleton) has no empty interior?