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A Review of General Topology. Part 6: Connectedness

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A Review of General Topology. Part 6: Connectedness A Review of General Topology. Part 6: Connectedness Wayne Aitken∗ March 2021 version This document is the sixth part in a series which gives a review of the basics of general document. This installment covers the concept of connectedness in general topology. Topics, such as compactness and topological groups will be covered in follow-up documents. There are several classes of readers that could benefit from this review. A reader who learned topology in the past but who has forgotten some details could use this as a summary of the key definitions and results. The proofs of many of the results are missing or are merely sketched, but enough details are given that a student comfortable with set-theoretic reasoning could supply the details. So a reader who has at least a causal familiarity of topology could use this series to systematically work through the subject, supplying the missing proofs along the way. The reader should be warned that this review is light on counter-examples and skips some less essential topics, so these notes are not a substitute for a more complete textbook. However, I have tried to hit all the really important elements. Can this series be used as a first introduction to general topology? I believe it can if used in conjunction with a knowledgeable instructor or knowledgeable friend, or if supplemented with other less concise sources that discuss additional examples and motivations. For the reader who wants to systematically work through the material with full proofs, I mention that is a rigorous account in the sense that it only relies on results that can be fully proved by the reader without too much trouble given the outlines provided here. The reader is expected to be versed in basic logical and set-theoretic techniques employed in the upper-division curriculum of a standard mathematics major. But other than that, the subject is self-contained.1 I have attempted to give full and clear statements of the definitions and results, with motivations provided where possible, and give indications of any proof that is not straightforward. However, my philosophy is that, at this level of mathematics, straightforward proofs are best worked out by the reader. So some of the proofs may be quite terse or missing altogether. Whenever a proof is not given, this signals to the reader that they should work out the proof, and that the proof is ∗Copyright c 2017{2021 by Wayne Aitken. Version of March 13, 2021. This work is made available under a Creative Commons Attribution 4.0 License. Readers may copy and redistribute this work under the terms of this license. 1Set theoretic reason here is taken to include not just ideas related to intersections, unions, and the empty set, but also complements, functions between arbitrary sets, images and preimages of functions, Cartesian products, relations such as order relations and equivalence relations, well- ordering and so on. 1 straightforward. Supplied proofs are sometimes just sketches, but I have attempted to be detailed enough that the prepared reader can supply the details without too much trouble. Even when a proof is provided, I encourage the reader to attempt a proof first before looking at the provided proof. Often the reader's proof will make more sense because it reflects their own viewpoint, and may even be more elegant. There are several examples included and most of these require the reader to work out various details, so they provide additional exercise. 1 Logical Dependencies This document assumes familiarity with some basic properties of R and its sub- field Q. For example the LUB property of R is critical to the material presented here. The related notions of Dedekind cuts and cut points is useful, but really only to motivate the definition of connected. The basic topological notions from the first part of this series are used exten- sively in this document. Facts about product topologies are also used. Results in the other earlier documents in the series, covering Hausdorff spaces, sequences, and metric spaces, are only needed for examples, if at all. 2 Connected Spaces The space of real numbers R has the property that any partition of R by two nonempty convex subsets S1 and S2 has a (unique) cut point x. The partition itself is called a Dedekind cut and the cut point is a point in the closure of both S1 and S2. The existence of cut points is traditionally described as a manifestation of the completeness of R, but in this section we view it as related to the general topological phenomenon of connectedness. Note: it turns out we can drop the convexity condition and replace it with the condition that S1 and S2 be nonempty; cut points will still exist although we may loose the uniqueness of the cut point if the sets are not convex. The notion of connection point is a generalization of the notion of cut point. Definition 1. Suppose S1 and S2 are two subsets of a topological space X.A point x 2 X is called a connection point for S1 and S2 if x 2 S1 and x 2 S2. If x 2 S the sometimes we say that x is a contact point of the subset S (this includes all limit points of S together with any point of S that is not a limit point of S). So a connection point for S1 and S2 is simply a point that is a contact point for both S1 and S2. We proved earlier that if x is a contact point of S in a topological space X, then the image f(x) is a contact point of f[S] in the space Y for any continuous function f : X ! Y . This implies that being a connection point is preserved by continuous functions as well: Proposition 1. Suppose f : X ! Y is continuous. If x 2 X is a connection point for subsets S1 and S2 then f(x) is a connection point for f[S1] and f[S2]. Intuitively we view a contact point as providing a \connection" or a \bridge" between two sets. This leads to the notion of connectedness: 2 Definition 2. A space X is said to be connected if, for all partitions of X by two nonempty subsets S1 and S2, there is a connection point for S1 and S2. Proposition 2. Suppose f : X ! Y is continuous and surjective. If X is connected then so is Y . Corollary 3. Suppose f : X ! Y is continuous then the image f[X] of a connected space X is a connected subspace of Y . Corollary 4. Suppose f : X ! Y is a homeomorphism. Then X is connected if and only if Y is connected. Proposition 5. Suppose a topological space X is partitioned by two sets S1 and S2. Then the following are equivalent: 1. There is no connection point for S1 and S2. 2. The subsets S1 and S2 are both closed in X. 3. The subsets S1 and S2 are both open in X. Proof. It is fairly straightforward to show (1) () (2) since S1 and S2 are disjoint. Similarly (2) () (3) is straightforward using complements. Proposition 6. Let X be a topological space. The following are equivalent: 1. X is connected. 2. X cannot be partitioned into two nonempty closed sets. 3. X cannot be partitioned into two nonempty open sets. Example 1. In a later section we will see that R is connected but Q is not. The empty space is connected, as is any singleton space. A discrete space with more than one point is not connected. Corollary 7. A space X is connected if and only if the only clopen subsets of X are X and ;. 3 Connected Subsets Now we consider the issue of connectedness for subsets of a fixed topological space X. A subset Y of a space X is said to be connected if Y is a connected space using the subspace topology. Recall that the closure operation is well-behaved with respect to the subspace topology in the following sense: if Y is a subspace of X and if S is a subset of Y then the closure of S in Y is equal to S \ Y where S is the closure of S in X. In other words, given a point y of Y and a subset S ⊆ Y , we have that y is a contact point of S in the topology of X if and only if y is a contact point of S in the subspace topology of Y . Since connection points are defined in terms of common contact points, we have the following: 3 Proposition 8. Suppose that Y is a subset of X and that S1 and S2 are subsets of Y . Suppose that y 2 Y . Then y is a connection point for S1 and S2 in the subspace topology on Y if and only if y is a connection point for S1 and S2 in the topology of X. We can restate Corollary 3 as follows: Proposition 9. The image of a connected subset under a continuous map is a connected subset of the codomain. Building on Proposition 6 we get the following: Proposition 10. Let Y be a connected subset of a topological space X. Suppose that A and B are open subsets of X such that (1) Y ⊆ A[B and (2) Y \A\B = ;. Then Y is a subset of either A or B. Similarly, suppose that A and B are closed subsets of X such that (1) Y ⊆ A[B and (2) Y \ A \ B = ;.
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