what is "Minimal Uncountable well-ordered set"? Can anyone make me understand what is "Minimal Uncountable well-ordered set" (Munkres, Topology, Example 2 of the limit point compactness section)?
I know what is Uncountablity and Well ordered set.
Thank You in Advance.
 A: Perhaps what you might be missing is a theorem of set theory which says (roughly speaking) that any set of well ordered sets is well ordered. 
To be precise, suppose that ${\cal A} =\{A_i\}_{i \in I}$ is a set, each of whose elements $A_i$ is a well ordered set, and let's suppose that no two elements of ${\cal A}$ are order isomorphic (i.e. for all $i \ne j \in I$ there does not exist an order preserving bijection between $A_i$ and $A_j$). Let's define a relation on ${\cal A}$, where $A_i  < A_j$ means that there exists an order preserving injection $f : A_i \to A_j$. 


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*Theorem: This relation is a well-ordering on ${\cal A}$. Therefore, ${\cal A}$ has a minimal element. 


So, to say that a well-ordered set $A$ is a "minimimal uncountable well ordered set" simply means that for any set ${\cal A}$ of uncountable well-ordered sets, no two of which are order isomorphic, if $A \in {\cal A}$ then $A$ is the minimal element of ${\cal A}$.
There are a few more theorems to ponder which help to iron out the understanding of this concept: 


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*Theorem: Suppose $A$ is an uncountable well ordered set, and suppose that for each $b \in A$ the set $\{a \in A \mid a < b\}$ is countable. Then $A$ is a minimal uncountable well ordered set. 

*Theorem: Any two minimal uncountable well ordered sets are order isomorphic. (So it is fair to speak about the minimal uncountable well ordered set, because it is unique up to order isomorphism)
For a more concrete example of this concept, the set of natural numbers $\mathbb{N}$ is the "minimal infinite well ordered set": for any set ${\cal A}$ of well ordered infinite sets (no two of which are order isomorphic), if $\mathbb{N} \in {\cal A}$ then $\mathbb{N}$ is the minimal element of ${\cal A}$. The other theorems mentioned have analogues here: 


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*Theorem: If $A$ is an infinite well ordered set, and if for each $b \in A$ the set $\{a \in A \mid a < b\}$ is finite, then $A$ is order isomorphic to $\mathbb{N}$.

*Theorem: Any minimal infinite well ordered set is order isomorphic to $\mathbb{N}$.
A: @cmi ℝ in the usual ordering is not well ordered. And if equipped with proper ordering, ℝ can be isomorphic to the "Minimal Uncountable well-ordered set" (assuming the continuum hypothesis), because under the continuum hypothesis, ℝ has the same cardinality as the "Minimal Uncountable well-ordered set", and therefore a bijection between them exists. We can then just define the ordering on ℝ based on the bijection and the ordering of the "Minimal Uncountable well-ordered set".
So the reason of ℝ (with the usual ordering) being "not well ordered minimal uncountable set" is not the reason you gave (can remove element), but the its ordering.
