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I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and  $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^{(1)}$ integer many times and not stop in the middle? Also, how do we prevent $f$ from oscillating infninitely? For example, if $X^{(1)}$ contains two edges $a,b$ with $a\cap b=\{(0,0)\}$, then $f(x)=(0,x^2\sin(1/x))$ can go infinitely many times into $a$ and $b$.

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and 
• $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^{(1)}$ integer many times and not stop in the middle? Also, how do we prevent $f$ from oscillating infninitely? For example, if $X^{(1)}$ contains two edges $a,b\subseteq\{0\}\times\mathbb{R}\subseteq\mathbb{R}^2$ with $a\cap b=\{(0,0)\}$, then $f(x)=(0,x^2\sin(1/x))$ can go infinitely many times into $a$ and $b$.

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^{(1)}$ integer many times and not stop in the middle? Also, how do we prevent $f$ from oscillating infninitely? For example, if $X^{(1)}$ contains two edges $a,b$, then $f(x)=(0,x^2\sin(1/x))$ can go infinitely many times into $a$ and $b$.

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^{(1)}$ integer many times and not stop in the middle? Also, how do we prevent $f$ from oscillating infninitely? For example, if $X^{(1)}$ contains two edges $a,b$ with $a\cap b=\{(0,0)\}$, then $f(x)=(0,x^2\sin(1/x))$ can go infinitely many times into $a$ and $b$.

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^(1)$ integer many times and not stop in the middle?

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.

I would be very grateful If someone could state a precise definition (direct one and inductive one) of a cell complex and CW-complex, since my intuition is telling that some restriction is missing and also, the definitions from several books seem to be very different. Conciseness is much desired, since everything seems so complicated.

1) DEFINITIONS From Introduction to Topological Manifolds (J. M. Lee)

and from Algebraic Topology (T. tom Dieck):

These definitions are so complicated, that I can't really see what's going on.

Let $\mathbb{B^n}$ denote a closed n-ball. As far as I know, a cell complex is a space, obtained as $X=\cup_{i\in\mathbb{N}_0} X^{(i)}$, such that

• $X^{(0)}$ is a discrete space and
• $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching $n$-cells, i.e. $X^{(n)}$ $=$ $X^{(n-1)}\cup_{f_\lambda}\coprod_{\lambda\in\Lambda}\mathbb{B^n}$ $=$ $X^{(n-1)}\coprod\coprod_{\lambda\in\Lambda}\mathbb{B^n}/_{x\sim f_\lambda(x);\; x\in\mathbb{S}^{n-1},\lambda\in\Lambda}$ and $A\subseteq X$ is closed in $X$ $\Longleftrightarrow$ $\forall n\in\mathbb{N}_0$: $A\cap X^{(n)}$ is closed in $X^{(n)}$.

2) MY PROBLEM: But shouldn't there be some condition on $f_\lambda$? For example, if we have a graph ($1$-dimensional complex) consisting of a single vertex and a single edge. Then when we are attaching $\mathbb{B^2}$, we can set $f$ to map the whole $S^1$ to a single point on the edge, that isn't the vertex. Thus we get a very weird space:

Shouldn't $f$ go along each loop/edge in $X^{(1)}$ integer many times and not stop in the middle? Also, how do we prevent $f$ from oscillating infninitely? For example, if $X^{(1)}$ contains two edges $a,b$, then $f(x)=(0,x^2\sin(1/x))$ can go infinitely many times into $a$ and $b$.

3) UNNECESSARY: If you have time/patience/interest, the definitions of a simplicial complex, abstract simplicial complex, Whitehead complex and any other complex are also welcome.