Show that the continuous image of a path-connected space is path-connected.

For open subset \(U\) of \(\mathbb{R^n}\) , show that \(U\) is connected if and only if \(U\) is path-connected.

Let \(X\) be a compact space with a nested sequence \(\{C_n\}\) of non-empty closed subsets: \(C_1 \supseteq C_2 \supseteq C_3 \cdots\) . Show that the intersection \(\cap_nC_n\) is non-empty.

A topological space \(X\) is compact if and only if for every collection \(C\) with finite intersection property, \(\cap_{C \in C}C\) is non-empty.

Let \(X\) be a first countable space and let \(A\) be a subset of \(X\). Then the following are true:

(1) (Sequence Lemma) \(A\) point \(x \in \overline{A}\) if and only if there is a sequence of points of A converging to \(x\).

(2) (Continuity Versus Sequential Continuity) Let \(f: X \rightarrow Y\) . Then \(f\) is continuous if and only if \(f\) is sequentially continuous.

A metric space \((X,d)\) is second countable if any one of the following holds true:

(1) \(X\) has a countable dense subset,

(2) \(X\) is compact.

Show that \(\mathbb{R_l}\) is not metrizable.

If \(X\) is second countable then every open covering of \(X\) contains a countable subcover.

Let \((X,d)\) be a metric space with metric. Let \(A\) be a subset of \(X\), and for \(x \in X\), let \(d(x,A):=inf\{d(x,a):a \in A \}\). Show that \(d(x,A)\) is a continuous function of \(x\).

Let \((X,\Omega )\) be a topological space with the property that any two disjoint non empty closed subsets of \(X\) can be separated by a continuous function. Show that for any disjoint non-empty closed subsets \(A\) and \(B\) of \(X\), there exist disjoint open sets \(U\) and \(V\) such that \(A \subseteq U\) and \(B \subseteq V\).

Consider the topological space \(\mathbb{R}_K\) . Show that there are no disjoint open sets \(U\) and \(V\) of \(\mathbb{R}_K\) such that \(\{0\} \subseteq U\) and \(K \subseteq V\). In particular, \(\mathbb{R}_K\) is Hausdorff but not normal.