Explore tens of thousands of sets crafted by our community.
Compactness and Connectedness
20
Flashcards
0/20
Cantor's Intersection Theorem
In a complete metric space, a nested sequence of non-empty, closed, and compact sets has a non-empty intersection.
Lebesgue's Number Lemma
Given an open cover of a compact metric space, there exists a positive number \(\delta\) (Lebesgue number) such that any subset of the space with diameter less than \(\delta\) is contained in some member of the cover.
Baire Category Theorem
In a complete metric space, the intersection of countably infinite dense open sets is dense.
Jordan Curve Theorem
In \(\mathbb{R}^2\), a simple closed curve separates the plane into an interior region which is bounded and an exterior region which is unbounded; both regions are connected.
Open Mapping Theorem
If \(f\) is a continuous surjection from a compact space to a Hausdorff space, then \(f\) is an open mapping. That is, it maps open sets to open sets.
Connected Component
The connected component of a point is the largest connected subset of the space that contains that point.
Urysohn's Lemma
For any two non-overlapping closed sets in a normal topological space, there is a continuous function on the space that takes on the value 0 on one set and 1 on the other.
Definition of Compactness
A subset K of a metric space is called compact if every open cover of K has a finite subcover.
Path Connectedness
A space is path connected if any two points can be connected by a continuous path within the space.
Bolzano-Weierstrass Theorem
Every bounded sequence in \(\mathbb{R}^n\) has a convergent subsequence.
Intermediate Value Theorem
If \(f: [a, b] \to \mathbb{R}\) is continuous and \(f(a)\) and \(f(b)\) have different signs, then there exists \(c\in [a,b]\) such that \(f(c) = 0\).
Equivalence of Compactness in Metric Spaces
In metric spaces, compactness, sequential compactness, and limit point compactness are equivalent.
Tietze Extension Theorem
If \(X\) is a normal space and \(A\) is a closed subset of \(X\), then every continuous function from \(A\) to \(\mathbb{R}\) can be extended to a continuous function on the whole space \(X\).
Heine-Borel Theorem
In Euclidean space \(\mathbb{R}^n\), a subset is compact if and only if it is closed and bounded.
Sequential Compactness
A space is sequentially compact if every sequence has a convergent subsequence.
Total Boundedness
A metric space is totally bounded if for every \(\epsilon > 0\), the space can be covered by a finite number of open balls of radius \(\epsilon\).
Locally Compact Spaces
A space is locally compact if every point has a compact neighborhood.
Sequences in Compact Sets
Every sequence in a compact set has a convergent subsequence with the limit also contained within the set.
Limit Point Compactness
A space is limit point compact if every infinite subset has a limit point within the space.
Definition of Connectedness
A space is said to be connected if it cannot be divided into two disjoint non-empty open sets.
© Hypatia.Tech. 2024 All rights reserved.
