Excluded Point Space is Locally Path-Connected

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Theorem

Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space.


Then $T$ is locally path-connected.


Proof

Consider the set $\mathcal B$ defined as:

$\mathcal B = \left\{{\left\{{x}\right\}: x \in S \setminus \left\{{p}\right\}}\right\} \cup \left\{{S}\right\}$

Then $\mathcal B$ is a basis for $T$.

Let $H \in \mathcal B$.

Then $\exists x \in S: H = \left\{{x}\right\}$.

From Point is Path-Connected to Itself we have that $H$ is path-connected.


Now consider the open set $S \in \mathcal B$.

Let $a, b \in S$.

Let $\mathbb I$ be the closed unit interval in $\R$.

Let $f: \mathbb I \to S$ be the mapping defined as:

$\forall x \in \mathbb I: f \left({x}\right) = \begin{cases} a & : x \in \left[{0 \,.\,.\, 1}\right) \\ b & : x = 1 \end{cases}$


Then $f^{-1} \left({S}\right) = \left[{0 \,.\,.\, 1}\right]$ which is open in $\mathbb I$ because $\left[{0 \,.\,.\, 1}\right] = \mathbb I$.


As $a$ and $b$ are any two points of $S$ it follows that $S$ is path-connected.


So we have shown that all elements of a basis of $T$ are path-connected.

Hence by definition $T$ is locally path-connected.

$\blacksquare$


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