query: $\tau$ be the topology on $\mathbb R$ generated by $\{(a,b] : a,b \in \mathbb R\}$ ; then is any basis for $(\mathbb R,\tau)$ must be uncountable ?
positive_passages.docid: 62793
positive_passages.text: $\mathbb{R}$ with the lower limit topology is not second-countable
negative_passages.docid: 116494
negative_passages.text: What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?
negative_passages.docid: 77957
negative_passages.text: How is $\mathbb N$ actually defined?
negative_passages.docid: 33207
negative_passages.text: Bases having countable subfamilies which are bases in second countable space
negative_passages.docid: 82989
negative_passages.text: T4 and first countable topology that is non metrizable
negative_passages.docid: 42652
negative_passages.text: Must $k$-subalgebra of $k[x]$ be finitely generated?
negative_passages.docid: 78412
negative_passages.text: Representing a countable field by $\Bbb N$ [transport of algebraic structure]
negative_passages.docid: 124864
negative_passages.text: A not free $\mathbb{Z}$-module
negative_passages.docid: 104486
negative_passages.text: Ideal in $\mathbb Z[x]$ which is not two-generated
negative_passages.docid: 132653
negative_passages.text: How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
negative_passages.docid: 124847
negative_passages.text: Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
negative_passages.docid: 108052
negative_passages.text: Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?
negative_passages.docid: 72190
negative_passages.text: Topological proof on discrete topology where $X$ is infinite
negative_passages.docid: 47251
negative_passages.text: Non-compactness of $\mathbb{R}$ with the cocountable topology
negative_passages.docid: 151950
negative_passages.text: Why is indiscrete topology unmetrizable?
negative_passages.docid: 55157
negative_passages.text: $\sigma$-algebras on a countable set is generated by partitions of the set
negative_passages.docid: 73690
negative_passages.text: If $\mathbb R^3\setminus V$ connected where $V$ is the subspace generated by $\{(1,1,1),(0,1,1)\} $