Section 11.6. Connected Topological Spaces - Faculty
... a continuous function is connected. So a topological space (X, T ) is connected if for each pair of points u, v ∈ X, there is a continuous map f : [0, 1] → X for which f (0) = u and f (1) = v. A topological space possessing this type of connectivity is called arcwise connected (or path connected). E ...
... a continuous function is connected. So a topological space (X, T ) is connected if for each pair of points u, v ∈ X, there is a continuous map f : [0, 1] → X for which f (0) = u and f (1) = v. A topological space possessing this type of connectivity is called arcwise connected (or path connected). E ...
2007 Spring final (2 hour)
... (a) (10pts.) Find an example of a Hausdorff but not regular space. (b) (10pts.) Find an example of a regular but not normal space. 2. Prove: (a) (20pts) A regular Lindelöf space is normal. (b) (20pts) A locally compact Hausdorff space is completely regular. (c) (20pts) The product space RJ is norma ...
... (a) (10pts.) Find an example of a Hausdorff but not regular space. (b) (10pts.) Find an example of a regular but not normal space. 2. Prove: (a) (20pts) A regular Lindelöf space is normal. (b) (20pts) A locally compact Hausdorff space is completely regular. (c) (20pts) The product space RJ is norma ...
ON SEMICONNECTED MAPPINGS OF TOPOLOGICAL SPACES 174
... (X, Tl) into (F, TJ) is semiconnected if/"1(A) is a closed and connected set in (X, 1L) whenever A is a closed and connected set in (Y, V). A mapping/ is bi-semiconnected if and only if/and/-1 are each semiconnected. Using the definition of G. T. Whyburn [5] a connected T+space (X, It) is said to be ...
... (X, Tl) into (F, TJ) is semiconnected if/"1(A) is a closed and connected set in (X, 1L) whenever A is a closed and connected set in (Y, V). A mapping/ is bi-semiconnected if and only if/and/-1 are each semiconnected. Using the definition of G. T. Whyburn [5] a connected T+space (X, It) is said to be ...
Midterm 1 solutions
... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...
... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...