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  1. The pasting lemma: A and B are closed sets. If f:A \to Y and g: B \to Y are both continuous, and f(x)=g(x) for all x \in A \cap B, then h: (A \cup B) \to Y defined by h(x)=f(x) for x \in A and h(x)=g(x) for x \in B is also continuous.
  2. Insight: Continuity (of f: X \to Y) does not only depend on the function itself, but also depend on the topologies on both X and Y. Indeed, with weaker topology on X or stronger topology on Y, there are less continuous functions from X to Y; on the contrast, with stronger topology on  X or weaker topology on Y, there are more continuous functions from X to Y.

How to measure the size of a set? It is easy for sets with finite elements, but how about sets with infinite elements? We have used ‘infinite’ several times in previous posts, as we all have a rough idea about what infinity means. In this post, we will give strict definition and other discussions concerning finite and infinite sets. Notice that \mathbb{N} has already been assumed.

1. Finite set

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When defining set, we mentioned that repetition and order are irrelevant for a set in the sense that \{a,b,c\}=\{b,c,a,b,b,c\}. This somewhat tells us the existence of order for the elements of sets. In fact, for many sets we are interested in, one or more orders are assumed in the common sense so that we can compare those elements. For example, for the set \{1,2,3,4\}, there is apparently an order called ‘less than’ (小于) which means we can ‘compare’ two elements in the manner whether one number is less than the other number or not. Similarly ‘less or equal to’ (小于等于), ‘greater than’ (大于), ‘greater or equal to’ (大于等于), and ‘equal to’ (等于) are other four orders for this set.

There is a mathematical way to define an ‘order’ for a set. This founds an important cornerstone for mathematical logic, and it is the main task of this post.

1. Ordered pair (有序对)

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In this post, we will study several important examples of set. The elements of these sets are all numbers. Notations are:

\mathbb{N}: Natural numbers.

\mathbb{Z}: Integers.

\mathbb{Q}: Rational numbers.

\mathbb{R}: Real numbers.

\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}.

Also, assume the addition (加法) and multiplication (乘法) of integers, i.e., 1+1=2, 3-5=-2, 4 \cdot (-2)=-8. Now we can start to develop these number systems (sets).

1. Natural numbers

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We deal with different objects in math such as numbers, functions(函数), operators(算子), and so on. It is important to indicate what kinds of objects we are dealing with before starting our study. Therefore, set plays a fundamental role in math. With different sets, we might have very different results for the same problem. For example, a quadratic equation(二次方程) ax^2+bx+c=0 with a \neq 0 and b^2-4ac<0 has no root(根) in \mathbb{R}(实数) , but two different roots in \mathbb{C}(虚数).

Here we only discuss naive set theory, which is presumed and used in most branches of mathematics. There is a much more strict set theory, which involves a lot of mathematical logic. Because of the reason we mentioned above, it has become the foundation of every other part of mathematics.

Set is one of basis conceptions(概念), which other definitions (定义) depend on. There is no strict (mathematical) way to define what is called a set. It is as if we have to define 1+1=2 and some other laws to define addition(加法). One could also choose to define 1+2=3 firstly, but anyway, there needs to be a initial conception.  Roughly speaking (大概地说), a set is a collection of objects (一些目标对象在一起的整体/汇合) we are interested in, which has specific properties.

1. Terminology and Notation 术语和记号

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