A continuous function does not always map open sets to open sets, but a continuous function will map compact sets to compact sets. ... etc.). For mere continuous most things have been mentioned: simple covering properties (variations on compactness, connectedness, Lindelöf) and separability. More complicated covering properties such as ...

Section 5.4 Continuous functions and Compact Sets. Now we're ready to see how continuous functions interact with compactness. Proposition 5.4.1.. Let \(f:A\to M\) be a continuous function from a subset of a metric space to a metric space. For any compact \(K\subseteq A\text{,}\) \(f_*(K)\subset M\) is compact.. Theorem 5.4.2. Max-Min Theorem. Let \(f:A\to \mathbb{R}\) be a continuous function ...

Lecture 18: Connectedness Now that we have properly de ned open, closed sets and limit points in a topological space, we can proceed to de ne the properties of connectedness and compactness for arbitrary topological space. The properties of connectedness and compactness are useful in proving three basic theorems about continuous functions

The next key observation is that continuous functions preserve compactness as well; a key difference is that it is not the inverse image of compact sets which must be compact, but the image itself. This is because of the "backwards" relationship between compactness and openness.

1 Continuous Functions on Compact Subsets of R 1.1 Preservation of Compactness and Extreme Value Theorem (EVT) Theorem 1 (Continuous Functions Preserve Compactness) If K∈K R and f∈C(R), then f(K) ∈K R. Proof 1 (via sequential compactness): (Theorem 4.4.1, Abbott) Let K∈ K R = Ks R and f∈C(R), and let us show that f(K) ∈K R = Ks R by ...

In the presence of completeness, continuous functions preserve totally bounded sets. Alternatively, we can strengthen the definition of boundedness even further to a property that is preserved by continuous functions; such a property is compactness, but it will emerge that compact sets are precisely the complete and totally bounded subsets. ...

Continuous functions preserve connectedness. Let \(f:A\to W\) be a continuous function from a subset of one normed space to another normed space. ... (T\) to be a continuous function. Furthermore, \(S^2\) is connected (that requires proof but we'll assume it for now).

6 Continuous functions preserve connectedness . . . . . . . . . . . . 8 ... restriction of f to C is the function g : C 7!B given by g(c) = f(c) for every c 2C. So ... the characterization of connectedness presented in the book. (As I explained before, the book does not talk about metric spaces, let alone about subsets ...

2 Continuous Functions on Connected Subsets of R 2.1 Preservation of Connected and Intermediate Value Theorem (IVT) Theorem 7 (Continuous Functions Preserve Connectedness) If A⊆R is connected and f∈C(A), then f(A) is connected. Proof: Suppose f(A) = B∪C where A,B ̸= ∅ and A∩B = ∅. We will use