Difference between revisions of "Topological property theorems"

Latest revision as of 08:37, 1 July 2015

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

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

Theorem	$X$ -Cmpct	$X$ -Cnctd	$X$ -Hsdrf	$\longrightarrow$	$f(X)$ -Cmpct	$f(X)$ -Cnctd	$f(X)$ -Hsdrf
Image of a connected set is connected	M	T	M	$\implies$	M	T	M
Image of a compact set is compact	T	M	M	$\implies$	T	M	M

Given a topological space, $(X,\mathcal{J})$ and a set $V\subseteq X$ then:

	Space properties		[Set properties			(relation)	Deduced properties]
Theorem	$X$ -Cmpct	$X$ -Hsdrf	$V$ -Open	$V$ -Clsd	$V$ -Cmpct	$\longrightarrow$	$V$ -Open	$V$ -Clsd	$V$ -Cmpct
Compact set in a Hausdorff space is closed	M	T	M		T	$\implies$	M	T	T (def)
Closed set in a compact space is compact	T	M	M	T		$\implies$	M	T (def)	T
Set in a compact Hausdorff space is compact iff it is closed	T	T	M		T	$\iff$	M	T	T (def)
Set in a compact Hausdorff space is compact iff it is closed	T	T	M	T		$\iff$	M	T (def)	T
Set in a compact Hausdorff space is compact iff it is closed	T	T	M	T	T $(\impliedby)$	$\iff$	M	T $(\impliedby)$	T

Here $\mathbb{R}$ is considered with the topology induced by the absolute value metric.

TODO: Formulate table

Theorems:

If $A\subseteq\mathbb{R}$ is compact $\implies$ $A$ is closed and bounded (page: Compact subset of the real line is closed and bounded)
The closed interval $[0,1]$ 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 $A$ 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

@@ Line 59: / Line 59: @@
 {{Ti key|ttmtt|[[Set in a compact Hausdorff space is compact iff it is closed]]}}{{Ti t}}{{Ti t}}{{Ti m}}{{Ti t}}{{Ti t|style="weak"|text={{M|(\impliedby)}}}} |{{Ti key|mtt|{{M|\iff}}}} {{Ti m}}{{Ti t|style=weak|text={{M|(\impliedby)}}}}{{Ti t}}
 |}
+==Real line==
+Here {{M|\mathbb{R} }} is considered with the topology induced by the [[Absolute value|absolute value]] metric.
+{{Todo|Formulate table}}
+'''Theorems:'''
+* If {{M|A\subseteq\mathbb{R} }} is compact {{M|\implies}} {{M|A}} is closed and bounded (page: [[Compact subset of the real line is closed and bounded]])
+* The closed interval {{M|[0,1]}} 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 {{M|A}} 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}}
 [[Category:Index]]
 [[Category:Topology]]