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I am trying to understand the Alexandroff compactification using Schubert's book. As far as I understand, the Alexandroff theorem can be stated as follows:

Theorem:(Alexandroff compactification) Let X be a locally compact space. Then, there exists a compact space $X^{*}$ such that $X$ is homeomorphic to a subspace $A$ of $X^{*}$ and $X^{*}-A=\{w\}$ (with $w$ a point of $X^{*}$ ). Also, $X^{*}$ is uniquely determined up to homeomorphism.

On the very first line of the proof, Schubert writes the following:

The set $X^*$ consists of the set $X$ and another point $w$.

I really don't get why he writes this: shouldn't it be $X^*=A\cup \{w\}$?

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    If you're exact you should say $X^\ast = X \cup \{w\}$, where $w \notin X$.Twice using $X$ reflects the idea that we want $X$ as the original space and $X$ as a subspace of a new space $X^\ast$ (to be defined) to be homeomorphic, using the identity in fact. Then $A =X$ in this construction. – 2017-02-04
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    But the unicity says that whenever we have a compact set $C$ such that for some point $w \in C$, we have $A:= C\setminus \{w\}$ is homeomorphic to $X$ ,then $C$ is homeomorphic to the $X^\ast$ we are now constructing. – 2017-02-04

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To show that any such compactification exists, you can define the underlying set of $ X^*$ as $ X\cup \{\omega \}$ for any $\omega\not\in X $ and then define a suitable topology on that set. To prove that this particular construction satisfies the properties stated in the theorem you will show that $ X $ is homeomorphic to the set $ A:=X\subset X^*$ (equipped with the subspace topology).

In short: What Schubert says is technically not true for any realization of $ X^*$! But it is true, by definition, for the obvious construction of a possible realization of $ X^*$. Also it is morally true for any realization of $ X^*$ since being homeomorphic to a subset is just as good as being that subset for all topological purposes

(Since I do not have the book I cannot judge whether Schubert actually claims that the equality holds or whether it is a definition in the sense of my first paragraph)