· Portal  Help Search Members Calendar 
Welcome Guest ( Log In  Register )  Resend Validation Email 
Join the millions that use us for their forum communities. Create your own forum today. Learn More · Register for Free  Welcome to Did You Know? 
JaneFairfax 
Posted: 03:36 Saturday 31 March 2007

The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 
Given a topological space X, the empty set and X itself are always “clopen” sets – sets that are both open and closed. (Note: We say that a subset of the topological space X is closed iff its complement in X is open.) If X has no other “clopen” sets, it is said to be connected. On the other hand, a topological space is said to be disconnected iff it is the disjoint union of two nonempty open sets. If a topological space Y is disconnected, then Y is the disjoint union of nonempty open sets S and T. Then S and T are closed (since they are each other’s complement), and they are also proper subsets of Y (since neither of them is empty). Hence Y, having “clopen” subsets other than the empty set and itself, is not connected. Conversely, if Y is not connected, then there exists a clopen set S which is neither the empty set nor Y. Since S is closed, its complement in Y is open, and since S is not the whole of Y, its complement in Y is not empty. Hence Y, being the disjoint union of S and its complement in Y, both of which are open and nonempty, is disconnected. Did you know? This proves that a topological space is disconnected if and only if it is not connected. 
algebraic topology 
Posted: 18:23 Thursday 17 June 2010

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? A topological space X is connected if and only if there exists no surjective continuous function from X onto the discrete twoelement space. 
JaneFairfax 
Posted: 17:08 Tuesday 03 August 2010

The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 
Did you know? A topological space is locally connected iff any element of any open subset belongs to a connected open set contained in that open subset. A subset of a topological space is a maximal connected subset iff it is connected and is not properly contained in any other connected subset. Any maximal connected subset of a locally connected topological space is open, and any topological space is the disjoint union of its maximal connected subsets. The cardinality of the class of all maximal connected subsets of a topological space is a topological invariant, meaning that it is preserved under homeomorphisms. 
algebraic topology 
Posted: 17:55 Tuesday 03 August 2010

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? In the language of category theory, there is a covariant functor from the category of topological spaces and continuous fuctions to the category of sets and set functions which maps each topological space to the set of its maximal connected subsets, with homeomorphisms in the first category corresponding to bijections in the other. 
JaneFairfax 
Posted: 09:53 Thursday 05 August 2010

The Enlightenment Group: Moderators Posts: 673 Member No.: 20 Joined: 03 Mar 2007 
Did you know? A more obvious – but far less interesting – covariant functor from the category of topological spaces and continuous functions to the category of sets and set functions is the one that maps each topological space to its underlying set. Such a functor is said to be forgetful because it “forgets” some of the structures of the first category! 
algebraic topology 
Posted: 10:55 Sunday 05 September 2010

Renaissance Group: Friends Posts: 143 Member No.: 14 Joined: 20 Feb 2007 
Did you know? The opposite of a forget functor is a free functor, e.g. the functor from the category of sets and set functions to the category of groups and group homomorphisms mapping each set to the free group generated by that set. 