The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

In fact, we can define the orbit space $C_{n}(X) := C_{\hat{n}}(X) / \Sigma_{n}$ as configuration space modded out by the symmetric group on $n$ elements $\Sigma_{n}$. Then it can be shown that $$\pi_{1}(C_{\hat{n}},\vec{p}) = PB_{n}, \; \text{and} \; \pi_{1}(C_{n},\tau(\vec{p})) = B_{n}$$ Where, $\tau: C_{\hat{n}}(X) \rightarrow C_{n}(X)$ is an $n!$-sheeted covering map called the orbit space projection, and $PB_{n}$ and $B_{n}$ are the pure braid group and braid group on $n$ strands, respectively. If this sounds interesting to you I suggest you read the following paper:

• Configuration Spaces and Braid Groups by Fred Cohen and Jonathan Pakianathan

The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

In fact, we can define the orbit space $C_{n}(X) := C_{\hat{n}}(X) / \Sigma_{n}$ as configuration space modded out by the symmetric group on $n$ elements $\Sigma_{n}$. Then it can be shown that $$\pi_{1}(C_{\hat{n}},\vec{p}) = PB_{n}, \; \text{and} \; \pi_{1}(C_{n},\tau(\vec{p})) = B_{n}$$ Where, $\tau: C_{\hat{n}}(X) \rightarrow C_{n}(X)$ is an $n!$-sheeted covering map called the orbit space projection, and $PB_{n}$ and $B_{n}$ are the pure braid group and braid group on $n$ strands, respectively. 

If this sounds interesting to you, then I suggest you read the following paper:

• Configuration Spaces and Braid Groups by Fred Cohen and Jonathan Pakianathan

The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

The configuration space of $n$ points in a topological space $X$ is usually defined to be,

$$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) \in X^{n} \; | \; z_{i} \neq z_{j} \; \text{if} \; i\neq j \}$$

A theorem which may be enlightening to your intuitive understanding of $C_{\hat{n}}(X)$ would be the following:

$$C_{\hat{n}}([0,1]) = \coprod_{i=1}^{n!} \Delta_{i}^{n}$$

Where $\coprod$ denotes disjoint union, and $\Delta_{i}^{n}$ denotes the $i^{th}$ copy of the $n$-simplex $\Delta^{n}$. We see that there are $n!$ $\Delta^{n}$'s because the symmetric group (which has order $n!$) acts freely on $C_{\hat{n}}(X)$, permuting the coordinates in each $\vec{z}=(z_{1},...,z_{n}) \in C_{\hat{n}}(X)$.

In fact, we can define the orbit space $C_{n}(X) := C_{\hat{n}}(X) / \Sigma_{n}$ as configuration space modded out by the symmetric group on $n$ elements $\Sigma_{n}$. Then it can be shown that $$\pi_{1}(C_{\hat{n}},\vec{p}) = PB_{n}, \; \text{and} \; \pi_{1}(C_{n},\tau(\vec{p})) = B_{n}$$ Where, $\tau: C_{\hat{n}}(X) \rightarrow C_{n}(X)$ is an $n!$-sheeted covering map called the orbit space projection, and $PB_{n}$ and $B_{n}$ are the pure braid group and braid group on $n$ strands, respectively. If this sounds interesting to you I suggest you read the following paper:

• Configuration Spaces and Braid Groups by Fred Cohen and Jonathan Pakianathan