MIDTERM EXAM
... Let Y = {a, b, c}. (a) Let p : X → Y be the function sending 1 7→ a, 2 7→ a, 3 7→ b, 4 7→ c. Find the quotient topology Tp on Y defined by the function p. (b) Let q : X → Y be the function sending 1 7→ a, 2 7→ b, 3 7→ b, 4 7→ c. Find the quotient topology Tq on Y defined by the function q. (c) Are t ...
... Let Y = {a, b, c}. (a) Let p : X → Y be the function sending 1 7→ a, 2 7→ a, 3 7→ b, 4 7→ c. Find the quotient topology Tp on Y defined by the function p. (b) Let q : X → Y be the function sending 1 7→ a, 2 7→ b, 3 7→ b, 4 7→ c. Find the quotient topology Tq on Y defined by the function q. (c) Are t ...
4. Connectedness 4.1 Connectedness Let d be the usual metric on
... (ii) If A is open in Y (in the subspace topology on Y ) and Y is an open subset of X then S is an open subset of X. (iii) If A is closed in Y (in the subspace topology on Y ) and Y is a closed subset of X then A is a closed subset of X. T (i) Suppose A is closed in Y (in the subspace topology). Then ...
... (ii) If A is open in Y (in the subspace topology on Y ) and Y is an open subset of X then S is an open subset of X. (iii) If A is closed in Y (in the subspace topology on Y ) and Y is a closed subset of X then A is a closed subset of X. T (i) Suppose A is closed in Y (in the subspace topology). Then ...
More on Semi-Urysohn Spaces
... upper plane is an example of a countable, connected, first countable Hausdorff space that fails to be semi-Urysohn. Any pair of nonempty regular closed sets has nonempty intersection. This shows that the irrational slope topology is not semi-Urysohn. We also note that the irrational slope topology i ...
... upper plane is an example of a countable, connected, first countable Hausdorff space that fails to be semi-Urysohn. Any pair of nonempty regular closed sets has nonempty intersection. This shows that the irrational slope topology is not semi-Urysohn. We also note that the irrational slope topology i ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.