# Topological property theorems

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## 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:

TODO: Mendelson - p165-167