Products and quotients via universal property
... This exercise consists of reinterpreting some of the results we’ve proved in class.
You may use results in the book without having to reprove them.
1. Let X and Y be topological spaces. Prove that X × Y has the following
universal property: if Z is a topological space and fX : Z → X and
fY : Z → Y a ...
... II. (a) Show that (−∞, a) ∪ [a, +∞) is a separation of the space R`
for any real a. (b) Describe connected components of R` and classify
all continuous maps R −→ R` .
Munkres exercise 3 on page 152.
Exercises on pages 157-158:
• Exercise 1 (imbeddings are defined on page 105)
• Exercise 2 (Hint: fir ...
... that f −1 is also continuous. We also say that two spaces are homeomorphic if
such a map exists.
If two topological spaces are homeomorphic, they are topologically equivalent
— using the techniques of topology, there is no way of distinguishing one space
from the other.
An autohomeomorphism (also kn ...
In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.