Space S000056

Smirnov's deleted sequence topology

Also known as: K-topology

Let $K = \{\frac{1}{n} : n \in \mathbb{N}\}$ . Smirnov's Deleted Sequence Topology is the topology on $\mathbb{R}$ consisting of all sets of the form $U \setminus B$ where $U \subset \mathbb{R}$ is open in the standard topology and $B \subset K$ .

Defined as counterexample #64 ("Smirnov's Deleted Sequence Topology") in DOI 10.1007/978-1-4612-6290-9.

Smirnov's deleted sequence topology is a counterexample to the converse of 4 theorems

Id	If	Then
T88	Path connected ∧ Has multiple points	¬Totally path disconnected
T164	Regular ∧ $\sigma$ -relatively-compact	$\sigma$ -compact
T486	Weakly locally compact ∧ Connected ∧ Locally connected ∧ $T_2$	$\sigma$ -connected
T541	Path connected	$\sigma$ -connected