CONTRA-CONTINUOUS FUNCTIONS AND STRONGLY S
... subfamily the closures of whose members cover X) THEOREM 2.10. The image of an almost compact space under contra-continuous, nearly continuous mapping is compact. PROOF. Let f (X, 7-) (Y, a) be contra-continuous and nearly continuous and let Xbe almost compact. Let (V,)zei be an open cover of Y. The ...
... subfamily the closures of whose members cover X) THEOREM 2.10. The image of an almost compact space under contra-continuous, nearly continuous mapping is compact. PROOF. Let f (X, 7-) (Y, a) be contra-continuous and nearly continuous and let Xbe almost compact. Let (V,)zei be an open cover of Y. The ...
Nonnormality of Cech-Stone remainders of topological groups
... But that remainder is not Lindelöf, simply observe that G is a P -space and that any P space with a Lindelöf remainder is discrete. We will show that the Čech-Stone remainder of this topological group G is not normal. Hence it is not true that the normality of a specific remainder implies that al ...
... But that remainder is not Lindelöf, simply observe that G is a P -space and that any P space with a Lindelöf remainder is discrete. We will show that the Čech-Stone remainder of this topological group G is not normal. Hence it is not true that the normality of a specific remainder implies that al ...
Over Chapter 1
... B. SCHOOL The table shows the enrollment of incoming freshmen at a high school over the last four years. The school wants to predict the number of freshmen for next year. Make a conjecture about the enrollment for next year. A. ...
... B. SCHOOL The table shows the enrollment of incoming freshmen at a high school over the last four years. The school wants to predict the number of freshmen for next year. Make a conjecture about the enrollment for next year. A. ...
Solutions to Homework 1
... with f ◦ s = 1Y . (Such a map is called a “section” of f . The fact that every surjection has a section is special to the category of sets: for example, if X and Y are topological spaces and f is continuous, there may not be a continuous s with this property.) Define h = g ◦ s, so h ◦ f = g ◦ s ◦ f ...
... with f ◦ s = 1Y . (Such a map is called a “section” of f . The fact that every surjection has a section is special to the category of sets: for example, if X and Y are topological spaces and f is continuous, there may not be a continuous s with this property.) Define h = g ◦ s, so h ◦ f = g ◦ s ◦ f ...
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.