# The property that compact sets in a space are closed lies strictly between T1 and T2

If a topological space^{} is Hausdorff^{} (${T}_{2}$), then every compact subset of that space is closed. If every compact subset of a space is closed, then (since singletons are always compact) then the space is accessible^{} (${T}_{1}$). There are spaces that are ${T}_{1}$ and have compact sets that are not closed, and there are spaces in which compact sets are always closed but that are not ${T}_{2}$.

Let $X$ be an infinite set^{} with the finite complement topology. Singletons are finite, and therefore closed, so $X$ is ${T}_{1}$. Let $S\beta \x8a\x82X$, and let $\mathrm{\pi \x9d\x94\xbd}$ be an open cover of $S$. Let $F\beta \x88\x88\mathrm{\pi \x9d\x94\xbd}$. Then $X\beta \x88\x96F$ is finite. Choosing a member of $\mathrm{\pi \x9d\x94\xbd}$ for each remaining element of $S$ shows that $\mathrm{\pi \x9d\x94\xbd}$ has a finite subcover. Thus, every subset of $X$ is compact. An infinite subset of $X$ will then be compact, but not closed.

Let $Y$ be an uncountable set with the countable complement topology. No two open sets are disjoint, so $Y$ is not Hausdorff. Let $C$ be a compact subset of $Y$. I shall show that $C$ is finite. Suppose $C$ is infinite, and let $S$ be an infinite sequence in $C$ without any repetitions. For any natural number^{} $n$, let ${U}_{n}$ be all the elements of $C$ except for all the ${S}_{k}$, where $k>n$. Then ${U}_{n}$ is open for each $n$, and $\{{U}_{n}\beta \x88\pounds n\beta \x88\x88\mathrm{\beta \x84\x95}\}$ covers $C$, but has no finite subset that covers $C$, contradicting the fact that $C$ is compact. This contradiction^{} arose by assuming a compact subset of $Y$ was infinite, all compact subsets of $Y$ are finite. $Y$ is ${T}_{1}$ (singleton sets are countable^{}), so all compact subsets of $Y$ are closed.

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Title | The property that compact sets in a space are closed lies strictly between T1 and T2 |
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Canonical name | ThePropertyThatCompactSetsInASpaceAreClosedLiesStrictlyBetweenT1AndT2 |

Date of creation | 2013-03-22 17:38:59 |

Last modified on | 2013-03-22 17:38:59 |

Owner | dfeuer (18434) |

Last modified by | dfeuer (18434) |

Numerical id | 5 |

Author | dfeuer (18434) |

Entry type | Result |

Classification | msc 54D30 |

Classification | msc 54D10 |

Related topic | T1Space |

Related topic | T2Space |

Related topic | Compact |