The image below (a Lemma and proof) is taken from Topology by James R. Munkres, 2nd Edition.
I understand the entirety of the proof up and till proving the statement: $$ F \text{ is a path homotopy} \implies \tilde{F} \text{ is a path homotopy}$$
Firstly Munkres states the set $p^{-1}(b_0)$ has the discrete topology as a subspace of $E$. What is meant by discrete topology in this case and how is this property deduced?
And secondly Munkres states that since $0 \times I$ is connected and $\tilde{F}$ is continuous, $\tilde{F}(0 \times I)$ is connected and must be a one point set. And then similarly for $\tilde{F}(1 \times I)$, thus proving $\tilde{F}$ is a path homotopy. How does one deduce this result from continuity and connectedness?
Many thanks, any help will be much appreciated.
A topological space $(X,\mathcal T)$ is discrete if every subset of $X$ is open, i.e., $\mathcal T=2^X.$
By the definition of a covering, the inverse image of a sufficiently small open neighbourhood of $b_0$ consists of disjoint open subsets of $E,$ each of which is mapped homeomorphically onto the original neighbourhood. In particular these homeomorphisms are bijective, so each of the disjoint open sets contains exactly one preimage of $b_0.$ Therefore the topology that $p^{-1}(b_0)$ inherits from $E$ is discrete, i.e., every one of its subsets is relatively open.
The continuous image of a connected topological space is connected.
It is not difficult to prove that any connected subspace of a discrete space is a singleton.