Timeline for Definition of Connected Subspace
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 5, 2010 at 9:15 | vote | accept | David Corwin | ||
| Jul 4, 2010 at 14:12 | answer | added | Tom Goodwillie | timeline score: 3 | |
| Jul 4, 2010 at 13:15 | comment | added | Karl Schwede | Davidac897, with regards to Munkres Lemma 23.1, which you quoted, I don't see the problem. You say that "Munkres shows that if a subspace Y of a space X is not connected, then there are two disjoint open subsets A, B such that the union of A and B contains Y." I don't see where Munkres says this. Munkres defines connected for topological spaces (not subsets), and then proves a lemma which gives a criterion for when a subspace of a topological space is connected. In the Lemma, $A$ and $B$ are not open! In Willie's example, ${a, c}$ is not connected in either the definition or the Lemma. | |
| Jul 4, 2010 at 13:09 | comment | added | Keenan Kidwell | A subspace $Y$ of a space $X$ is connected if it is connected in the subspace topology, i.e., if there do not exist disjoint open subsets $U$ and $V$ of $Y$ such that $Y=U\cup V$. Munkres is giving an alternative characterization of this definition. | |
| Jul 4, 2010 at 13:09 | answer | added | Abhishek Parab | timeline score: 2 | |
| Jul 4, 2010 at 12:58 | answer | added | Willie Wong | timeline score: 3 | |
| Jul 4, 2010 at 12:43 | comment | added | David Corwin | My question is: what is the standard definition of connected component? | |
| Jul 4, 2010 at 12:37 | comment | added | Abhishek Parab | Please clarify your question. Is it that you have a definition in mind, perhaps more natural than what Munkres says? | |
| Jul 4, 2010 at 12:36 | comment | added | David Corwin | So in Willie's example, is {a,c} connected or not? What is the connected component of the point a in the space {a,b,c}? | |
| Jul 4, 2010 at 12:34 | comment | added | David Corwin | Regardless of Munkres, my question is: when people say Y is a connected subset, do they always mean simply that it is a connected subspace? In this what is used in the definition of connected component? (Connected component is the maximal connected subset containing a point, and two points are in the same connected component iff there is a connected subset containing both.) Or do people ever mean simply mean by "connected subset" that there are no disjoint open subsets of $X$ satisfying my conditions. | |
| Jul 4, 2010 at 12:34 | comment | added | David Corwin | Your statement is exactly why I'm confused. Here is what Munkres writes, word-for-word: "If Y is subspace of X, a separation of Y is a pair of disjoint nonempty sets A and B whose union is Y, neither of which contains a limit point of the other. The space Y is connected if there exists no separation of Y." I assume he means "disjoint" in X. | |
| Jul 4, 2010 at 12:32 | comment | added | Willie Wong | For the few of us that don't have Munkres handy, can you provide the relevant statements and definitions for your first sentence? In particular, I am thinking about the space X = {a,b,c} and Y = {a,c} with topology on X generated by {a,b} and {b,c}. So as you stated it, that can't possibly be a theorem (but you said "Munkres showed"). | |
| Jul 4, 2010 at 12:31 | comment | added | José Figueroa-O'Farrill | Aren't you missing the condition that the closure of $A$ should not intersect $B$ and viceversa? | |
| Jul 4, 2010 at 12:24 | comment | added | Robin Chapman | There are examples of topological spaces $X$ and a non-connected subspace $Y$ such that there are no disjoint subspaces $A$ and $B$ of $X$ satisfying your conditions. Are you sure that's what Munkres actually says? | |
| Jul 4, 2010 at 12:16 | history | asked | David Corwin | CC BY-SA 2.5 |