Let f: X →Y be a continuous function from a topological...

Let $f: X \rightarrow Y$ be a continuous function from a topological space $X$ to a topological space $Y$. (a) Prove that the image $f(C)$ of a connected set $C$ is connected and that the image $f(K)$ of a compact set $K$ is compact. (b) Using a continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$, provide examples illustrating that the inverse image $f^{-1}(C)$ of a connected set $C$ need not be connected and that the inverse image $f^{-1}(K)$ of a compact set need not be compact.

Let $f: X \rightarrow Y$ be a continuous function from a topological space $X$ to a topological space $Y$.
(a) Prove that the image $f(C)$ of a connected set $C$ is connected and that the image $f(K)$ of a compact set $K$ is compact.
(b) Using a continuous function $f: \mathbf{R} \rightarrow \mathbf{R}$, provide examples illustrating that the inverse image $f^{-1}(C)$ of a connected set $C$ need not be connected and that the inverse image $f^{-1}(K)$ of a compact set need not be compact.

Key Concepts

Continuous Functions

A continuous function between topological spaces is one that maintains the structure of openness; that is, the preimage of every open set in the codomain is an open set in the domain. This concept is crucial because many topological properties, such as compactness and connectedness, behave predictably under continuous mappings.

Connectedness

Connectedness is a property of a space that implies it cannot be divided into two disjoint, nonempty open subsets. This concept is fundamental in topology because it ensures that a continuous image of a connected set remains connected, preserving the 'wholeness' of the set.

Compactness

Compactness generalizes the idea of closed and bounded sets, requiring that every open cover of the set has a finite subcover. In topology, continuous functions are known to preserve compactness, meaning that the image of a compact set under a continuous function is also compact, which is vital for various convergence and limit arguments.

Image and Inverse Image

The image of a set under a function comprises all the output points corresponding to the inputs from that set, while the inverse image consists of all points in the domain that map into a given subset of the codomain. These concepts are particularly significant in topology because they provide the framework for understanding how properties like connectedness and compactness transform under continuous mappings.

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