Topological property theorems

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, $(X,\mathcal{J})$ 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

Properties of a set in a space

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

Real line

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

---