Compact - Revision history

Kfagerstrom at 22:50, 21 May 2023

2023-05-21T22:50:19Z

2023-05-21T22:49:55Z

2023-05-21T22:22:23Z

2022-12-07T19:36:30Z

2022-12-07T19:06:44Z

2022-12-07T19:06:26Z

2022-12-07T19:05:58Z

Created page with " (EVT = Extreme Value Theorem):"

New page

(EVT = Extreme Value Theorem):

← Older revision		Revision as of 22:50, 21 May 2023
Line 15:		Line 15:

	Let $$A$$ be a subspace of a topological space $$(X,𝒯_X)$$. The space $$A$$ is compact if and only if for every collection $$𝒞$$ of open sets in $$X$$ satisfying $$A ⊆ ∪_{C ∈ 𝒞} C$$, there is a finite subcollection $$𝒟 ⊆ 𝒞$$ such that $$A ⊆ ∪_{D ∈ 𝒟} D$$.		Let $$A$$ be a subspace of a topological space $$(X,𝒯_X)$$. The space $$A$$ is compact if and only if for every collection $$𝒞$$ of open sets in $$X$$ satisfying $$A ⊆ ∪_{C ∈ 𝒞} C$$, there is a finite subcollection $$𝒟 ⊆ 𝒞$$ such that $$A ⊆ ∪_{D ∈ 𝒟} D$$.
		+
		+	[[Category:Point-Set Topology]]

@@ Line 7: / Line 7: @@
 If $$X$$ is a compact space and $$X/∼$$ is a quotient space, then $$X/∼$$ is compact.
-If $$A$$ is a subspace of a compact space $$X and $$A$$ is a closed subset in $$X$$, then $$A$$ is compact.
+If $$A$$ is a subspace of a compact space $$X$$ and $$A$$ is a closed subset in $$X$$, then $$A$$ is compact.
 (EVT = Extreme Value Theorem) Let $$X$$ be a compact space, and let $$f: X → (ℝ,𝒯_{\rm Eucl})$$ be a continuous function. Then there exist $$c,d ∈ X$$ such that for all $$p ∈ X, f(c) ≤ f(p) ≤ f(d)$$.

← Older revision		Revision as of 22:22, 21 May 2023
Line 1:		Line 1:
	=Compact (Topology)=		=Compact (Topology)=
−	~~(EVT = Extreme Value Theorem)~~

−	A continuous image of a compact space is compact. That is, if X,Y are topological spaces, and if X is compact and f: X → Y is a continuous surjective function, then Y is compact.	+	A topological space $$X$$ is '''compact'''' if every open covering of $$X$$ contains a finite subcollection that also covers $$X$$.
		+
		+	Compactness is a homeomorphism invariant.
		+
		+	If $$X$$ is a compact space and $$X/∼$$ is a quotient space, then $$X/∼$$ is compact.
		+
		+	If $$A$$ is a subspace of a compact space $$X and $$A$$ is a closed subset in $$X$$, then $$A$$ is compact.
		+
		+	(EVT = Extreme Value Theorem) Let $$X$$ be a compact space, and let $$f: X → (ℝ,𝒯_{\rm Eucl})$$ be a continuous function. Then there exist $$c,d ∈ X$$ such that for all $$p ∈ X, f(c) ≤ f(p) ≤ f(d)$$.
		+
		+	A continuous image of a compact space is compact. That is, if $$X,Y$$ are topological spaces, and if $$X$$ is compact and $$f: X → Y$$ is a continuous surjective function, then $$Y$$ is compact.
		+
		+
		+	Let $$A$$ be a subspace of a topological space $$(X,𝒯_X)$$. The space $$A$$ is compact if and only if for every collection $$𝒞$$ of open sets in $$X$$ satisfying $$A ⊆ ∪_{C ∈ 𝒞} C$$, there is a finite subcollection $$𝒟 ⊆ 𝒞$$ such that $$A ⊆ ∪_{D ∈ 𝒟} D$$.

@@ Line 1: / Line 1: @@
+=Compact (Topology)=
 (EVT = Extreme Value Theorem)
 A continuous image of a compact space is compact. That is, if X,Y are topological spaces, and if X is compact and f: X → Y is a continuous surjective function, then Y is compact.

← Older revision		Revision as of 19:06, 7 December 2022
Line 1:		Line 1:
−	(EVT = Extreme Value Theorem)	+
		+	(EVT = Extreme Value Theorem)

	A continuous image of a compact space is compact. That is, if X,Y are topological spaces, and if X is compact and f: X → Y is a continuous surjective function, then Y is compact.		A continuous image of a compact space is compact. That is, if X,Y are topological spaces, and if X is compact and f: X → Y is a continuous surjective function, then Y is compact.