Irrational Slope Topology

  5
T0 1
T1 1
Hausdorff 1
T3 0
T4 •
T5 •
Completely Hausdorff 0
T3.5 0
Urysohn 0
Semiregular 0
Perfectly Normal •
Separable •
Second Countable •
First Countable •
Lindelof •
Compact •
Sigma-Compact •
Countably Compact •
Sequentially Compact •
Limit Point Compact •
Pseudocompact •
Locally Compact •
Strongly Locally Compact •
Connected •
Path Connected •
Arc Connected •
Locally Connected •
Locally Path Connected •
Metrizable •
Totally Pathwise Disconnected •
Totally Disconnected •
Totally Separated •

Description: Let $X = { (x,y) in mathbb{Q} imes mathbb{Q} : y geq 0 }$ and fix some irational number $ heta$. The irrational slope topology on $X$ is generated by $epsilon$-neighborhoods of the form $N_epsilon(x,y)$ = { (x,y) } cup B_epsilon(x + y/ heta) cup B_epsilon(x - y/ heta)$ where $B_epsilon(zeta) = { r in mathbb{Q}: |r-zeta| < epsilon }$, $mathbb{Q}$ being the rationals on the $x$-axis. Each $N_epsilon(x,y)$ consists of ${(x,y)}$ plus two intervals on the rational $x$-axis centered at the two irrational points $x pm y/ heta$; the lines joining these points to $(x,y)$ have slope $pm\theta$.

Human-entered properties: Computer-inferred properties: Deleted properties: (none)

Sometimes you'll see a check properties link on the left column of this page. If you see it then it means the computer has not checked the human entered properties. The check properties link invokes the logical inference engine to determine if there are any properties which can be determined automatically.