# Topological property theorems

From Maths

## Contents

## Using this page

This page is an index for the various theorems involving topological properties, like compactness, connectedness, so forth.

TODO: Document this

The a few types of theorems are (like):

- Image of a compact space is compact
- Notice this is given X is compact, then Y is compact

- A continuous and bijective function from a compact space to a Hausdorff space is a homeomorphism
- Notice this is given X is compact, Y is Hausdorff, f bijective THEN homeomorphism

- A closed set in a compact space is compact
- Given a set, closed, X compact then set compact

## Properties carried forward by continuity

Given two topological spaces, [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](Y,\mathcal{K})[/ilmath] and a map, [ilmath]f:X\rightarrow Y[/ilmath] that is continuous then:

Theorem | [ilmath]X[/ilmath]-Cmpct | [ilmath]X[/ilmath]-Cnctd | [ilmath]X[/ilmath]-Hsdrf | [ilmath]\longrightarrow[/ilmath] | [ilmath]f(X)[/ilmath]-Cmpct | [ilmath]f(X)[/ilmath]-Cnctd | [ilmath]f(X)[/ilmath]-Hsdrf |
---|---|---|---|---|---|---|---|

Image of a connected set is connected | M | T | M | [ilmath]\implies[/ilmath] | M | T | M |

Image of a compact set is compact | T | M | M | [ilmath]\implies[/ilmath] | T | M | M |

## Properties of a set in a space

Given a topological space, [ilmath](X,\mathcal{J})[/ilmath] and a set [ilmath]V\subseteq X[/ilmath] then:

Space properties | [Set properties | (relation) | Deduced properties] | ||||||
---|---|---|---|---|---|---|---|---|---|

Theorem | [ilmath]X[/ilmath]-Cmpct | [ilmath]X[/ilmath]-Hsdrf | [ilmath]V[/ilmath]-Open | [ilmath]V[/ilmath]-Clsd | [ilmath]V[/ilmath]-Cmpct | [ilmath]\longrightarrow[/ilmath] | [ilmath]V[/ilmath]-Open | [ilmath]V[/ilmath]-Clsd | [ilmath]V[/ilmath]-Cmpct |

Compact set in a Hausdorff space is closed | M | T | M | T | [ilmath]\implies[/ilmath] | M | T | T (def) | |

Closed set in a compact space is compact | T | M | M | T | [ilmath]\implies[/ilmath] | M | T (def) | T | |

Set in a compact Hausdorff space is compact iff it is closed | T | T | M | T | [ilmath]\iff[/ilmath] | M | T | T (def) | |

Set in a compact Hausdorff space is compact iff it is closed | T | T | M | T | [ilmath]\iff[/ilmath] | M | T (def) | T | |

Set in a compact Hausdorff space is compact iff it is closed | T | T | M | T | T [ilmath](\impliedby)[/ilmath] | [ilmath]\iff[/ilmath] | M | T [ilmath](\impliedby)[/ilmath] | T |

## Real line

Here [ilmath]\mathbb{R} [/ilmath] is considered with the topology induced by the absolute value metric.

TODO: Formulate table

**Theorems:**

- If [ilmath]A\subseteq\mathbb{R} [/ilmath] is compact [ilmath]\implies[/ilmath] [ilmath]A[/ilmath] is closed and bounded (page: Compact subset of the real line is closed and bounded)
- The closed interval [ilmath][0,1][/ilmath] is compact Closed unit interval of real line is compact
- Each closed interval of the real line is compact Closed interval of the real line is compact
- A subset [ilmath]A[/ilmath] of the real line is compact
*if and only if*it is closed and bounded Subset of real line is compact if and only if it is closed and bounded

TODO: Mendelson - p165-167