Find the \(14^{th}\) Lecture/tutorial notes linked: MA 406 Lecture Notes (1-14), (Google Drive Link)

Connectedness

• Components of a topological space \(X\) emerge as the equivalence class of the relation \(x \sim y\) if there exists a connected subset of \(X\) containing both \(x\) and \(y\). We see that this indeed guarantees that any connected subset of \(X\) intersects at most one component and that each component is itself connected.

Compactness

• A cover of \(X\) is a collection of subsets of \(X\) such that its union is \(X\). If all the subsets in the cover are open, it is called an open cover. For a subset \(Y\) of \(X\), collection of subsets of \(X\) is said to cover \(Y\) if the union is its elements contains \(Y\).
• A space \(X\) is said to be compact if every open cover contains a finite subcover. A subset \(Y\) of $X\(is compact if and only if every covering of\)Y\(by open sets in\)X\(contains a finite subcovering of\)Y$$.
• Every closed subset of a compact space is compact. Every compact subset of a Hausdorff space is closed.
• Next time we shall show that if \(X\) is compact and \(Y\) is compact then the product \(X \times Y\) is compact. This will use tubular neighbourhoods and basic open sets to cover and then extract tubular neighbourhoods. Cover \(X\) by the tubular neighborhood. For infinitely many spaces, Tychnoff’s theorem which will be covered later.