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2 changes: 1 addition & 1 deletion spaces/S000099/README.md
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name: Counterexamples in Topology
---
Let $X$ be the unit square $[0,1] \times [0,1] \subset \mathbb{R}^2$. For a point $(x,y)$ not on the diagonal (for $x \ne y$), basic neighborhoods are given by the vertical segments $\{x\} \times (y - \varepsilon,y + \varepsilon)$ for $\varepsilon > 0$. For a point $(x,x)$ on the diagonal, basic open neighborhoods are given by open horizontal strips minus a finite number of vertical lines, i.e., sets of the form $S \times (x - \varepsilon,x + \varepsilon)$ for $\varepsilon > 0$ and $S$ cofinite subset of $[0,1]$ not containing $x$.
Let $X$ be the unit square $[0,1] \times [0,1] \subset \mathbb{R}^2$. For a point $(x,y)$ not on the diagonal (for $x \ne y$), basic neighborhoods are given by the vertical segments $\{x\} \times (y - \varepsilon,y + \varepsilon)$ for $\varepsilon > 0$. For a point $(x,x)$ on the diagonal, basic open neighborhoods are given by open horizontal strips minus a finite number of vertical lines, i.e., sets of the form $S \times (x - \varepsilon,x + \varepsilon)$ for $\varepsilon > 0$ and $S$ cofinite subset of $[0,1]$ containing $x$.

Every vertical slice is homeomorphic to {S158}. Every horizontal slice is homeomorphic to {S154}, with the homeomorphism sending the point on the diagonal to $\infty$. The diagonal $\{(x,x) | x \in [0, 1]\}$ is homeomorphic to {S158}.

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