“Connection gives purpose and meaning to our lives.”
―Brené Brown
Subsection3.3.1Connected
Motivation: Characterize spaces for which the Intermediate Value Theorem holds.
Definition3.3.1.
A topological space \(X\) is connected if whenever \(U\) and \(V\) are open sets in \(X\) satisfying \(U \cup V = X\) and \(U \cap V = \es\text{,}\) then either \(U = \es\) or \(V = \es\text{.}\)
Definition3.3.2.
A disconnection of a space \(X\) is a pair \(U,V\) of nonempty disjoint open sets in \(X\) whose union is \(X\text{.}\)
Theorem3.3.3.IVT = Intermediate Value Theorem.
Let \(X\) be a connected space, and let \(f: X \to (\R,\cT_{\text{Eucl}})\) be a continuous function. If \(p,q \in X\) and \(r \in \R\) satifies \(f(p) < r < f(q)\text{,}\) then there is an \(x \in X\) with \(f(x)=r\text{.}\)
Example3.3.4.
Examples
Theorem3.3.5.
A space \(X\) is connected if and only if the subsets of \(X\) that are both open and closed in \(X\) are \(X\) and \(\es\text{.}\)
Interactions with constructions and continuous functions:.
Theorem3.3.6.
A continuous image of a connected space is connected. That is, if \(X,Y\) are topological spaces, and if \(X\) is connected and \(f: X \to Y\) is a continuous function, then the subspace \(f(X)\) of \(Y\) is connected.
Corollary3.3.7.
Connectedness is a homeomorphism invariant.
Corollary3.3.8.
If \(X\) is a connected space and \(X/\sim\) is a quotient space, then \(X/\sim\) is connected.
Theorem3.3.9.
If \(X\) and \(Y\) are connected spaces, then the product space \(X \times Y\) is connected.
Remark3.3.10.
Connectedness is not preserved by subspaces or continuous preimages.
Lemma3.3.11.
Let \(X\) be a topological space and suppose that \(X = \cup X_\alpha and \cap X_\alpha \ne \es\text{.}\) If \(X_\alpha\) is a connected subspace of \(X\) for all \(\alpha\text{,}\) then \(X\) is connected.
Proposition3.3.12.
If \(Y\) is a connected subspace of a space \(X\) and if \(U,V\) are a disconnection of \(X\text{,}\) then either \(Y \sse U\) or \(Y \sse V\text{.}\)
Connectedness and Euclidean topology:.
Theorem3.3.13.
A subspace \(Y\) of \((\R,\cT_{\text{Eucl}})\) is connected if and only if \(Y\) is either an interval, a ray, or \(\R\text{.}\)
Proposition3.3.14.
\(\R\) and \(\R^n\) (with \(n \geq 2\)) with the Euclidean topology are not homeomorphic.
Theorem3.3.15.
Let \(X\) and \(Y\) be topological spaces with \(X \cap Y = \es\text{,}\) and let \(x_0 \in X\) and \(y_0 \in Y\text{.}\) Let \(Z := X \cup Y\) have the disjoint union topology. Define \(\sim\) to be the minimal equivalence relation on \(Z\) such that \(x_0 \sim y_0\text{,}\) and let \(Z/\sim\) be the corresponding quotient space. Then:
There are embeddings \(X \to Z/\sim\) and \(Y \to Z/\sim\text{.}\)
If \(X\) and \(Y\) are connected, then \(Z/\sim\) is connected.
Subsection3.3.2Path connected
Convention3.3.16.
Let \(I\) denote the topological space \([0,1]\) with the Euclidean subspace topology.
Definition3.3.17.
Let \(X\) be a topological space and let \(p,q \in X\text{.}\) A path in \(X\) from \(p\) to \(q\) is a continuous function \(f: I \to X\) such that \(f(0) = p\) and \(f(1) = q\text{.}\)
Definition3.3.18.
A space \(X\) is path-connected, or PC, if for all \(p,q \in X\text{,}\) there is a continuous function \(f: I \to X\) such that \(f(0) = p\) and \(f(1) = q\) (that is, there is a path from \(p\) to \(q\)).
Example3.3.19.
Examples
Interactions with constructions and continuous functions:.
Theorem3.3.20.
A continuous image of a path-connected space is path-connected.
Corollary3.3.21.
Path-connectedness is a homeomorphism invariant.
Theorem3.3.22.
If \(X_\alpha\) is a path-connected space for all \(\alpha\text{,}\) then the product space \(\prod_\alpha X_\alpha\) is path-connected.
If \(X\) is a path-connected space and \(\sim\) is an equivalence relation on \(X\text{,}\) then the quotient space \(X/\sim\) is path-connected.
Remark3.3.23.
Path-connectedness is not preserved by subspaces or continuous preimages.
Lemma3.3.24.
Let \(X\) be a topological space and suppose that \(X = \cup X_\alpha\) and \(\cap X_\alpha \ne \es\text{.}\) If \(X_\alpha\) is PC subspace of \(X\) for all \(\alpha\text{,}\) then \(X\) is PC.
Interactions with homeomorphism invariants:.
Theorem3.3.25.
If \(X\) is a path-connected space, then \(X\) is connected.
Definition3.3.26.
The flea and comb space is the subset \(X := \{(0,1)\} \cup (I \times \{0\}) \cup (\cup n \in \N \{1/n\} \times I)\) of\(\R^2\) with the Euclidean subspace topology. The point \((0,1)\) is the flea and the subspace \(X\setminus\{(0,1)\}\) is the comb.
Theorem3.3.27.
Connectedness does not imply path-connectedness. In particular, the flea-and-comb space is connected but not path-connected.
Theorem3.3.28.
A subspace \(Y\) of \((\R,\cT_{\text{Eucl}})\) is path-connected iff \(Y\) is either an interval, ray, or \(\R\text{.}\)
Subsection3.3.3Components
Definition3.3.29.
Let \(X\) be a topological space, and let \(\sim_{cc}\) be the equivalence relation on \(X\) defined by \(p \sim_{cc} q\) if and only if there is a connected subspace \(A\) of \(X\) with \(p,q \in A\text{.}\) A connected component of \(A\) is an equivalence class for the relation \(\sim_{cc}\text{.}\)
Definition3.3.30.
Let \(X\) be a topological space, and let \(\sim_{pc}\) be the equivalence relation on \(X\) defined by \(p \sim_{pc} q\) if and only if there is a path in \(X\) from \(p\) to \(q\text{.}\) A path component of \(A\) is an equivalence class for the relation \(\sim_{pc}\text{.}\)
Example3.3.31.
Examples
Lemma3.3.32.
\(\sim_{cc}\) and \(\sim_{pc}\) are equivalence relations.
Theorem3.3.33.
Let \(X\) be a topological space. Then \(X\) is a disjoint union of its connected components. Moreover, each connected component is a disjoint union of path components.
Theorem3.3.34.
The number (cardinality) of connected components is a homeomorphism invariant.
The number (cardinality) of path components is a homeomorphism invariant.
