Difference between revisions of "Topological property theorems"
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{{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}} | {{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:Index]] | ||
[[Category:Topology]] | [[Category:Topology]] | ||
Latest revision as of 08:37, 1 July 2015
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
