3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Public sphere pedagogy opinion through critical rational discourse. Habermas argued that these discussions served to fill the gap between the state and the people by creating what he called a "civil society". Further, Habermas argued that a functioning public sphere was critical to maintaining a healthy democratic order and deliberative democracy. This traditional notion of a face-to-face public sphere has evolved with the invention of online technologies. The public sphere no longer requires a physical setting, but can manifest in cyberspace. The coffee house discussions idealized by Habermas have expanded to blogs, discussion forums, and online videos. Henry Giroux in particular discusses the