Topological property theorems

Using this page

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

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TODO: Document this

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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.

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TODO: Formulate table

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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

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TODO: Mendelson - p165-167

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