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I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

(Here endeth the list of assumptions.)

Then $?$ is a generalized homology vanishing in negative domensions and vanishing on a point (therefore on CW complexes) but not vanishing on the path-connected thing that you get by attaching a sutiable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$

I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

(End of list of assumed properties.)

Then $?$ is a generalized homology theory vanishing in negative dimensions and vanishing on a point (therefore on CW complexes) but not vanishing on the path-connected space that you get by attaching a suitable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$

I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

(Here endeth the list of assumptions.)

Then $?$ is a generalized homology vanishing in negative domensions and vanishing on point (therefore on CW complexes) but not vanishing on the path-connected thing that you get by attaching a sutiable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$

I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

(Here endeth the list of assumptions.)

Then $?$ is a generalized homology vanishing in negative domensions and vanishing on a point (therefore on CW complexes) but not vanishing on the path-connected thing that you get by attaching a sutiable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$

I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

Then $?$ is a generalized homology vanishing in negative domensions and vanishing on point (therefore on CW complexes) but not vanishing on the path-connected thing that you get by attaching a sutiable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$

I'm not sure exactly what Cech homology is, but I'll assume that something which may or may not be called Cech homology has the following properties:

It's a generalized homology theory.

It vanishes in negative dimensions.

It satisfies the dimension axiom (so it's a homology theory).

There's a natural map from singular to Cech that is part of a natural triangle, an exact sequence

$\dots \to Cech_{n+1}\to ?_n\to Sing_n\to Cech_n\to \dots$.

The map $Sing_0\to Cech_0$ is always surjective, so that $?_n$ vanishes for $n<0$.

The map $Sing_0(TSC)\to Cech_0(TSC)$ is not injective if TSC is the topologist's sine curve.

(Here endeth the list of assumptions.)

Then $?$ is a generalized homology vanishing in negative domensions and vanishing on point (therefore on CW complexes) but not vanishing on the path-connected thing that you get by attaching a sutiable $1$-cell to TSC.

The direct sum of ? and singular will then do the job, because $?_0(TSC\cup cell)=?_0(TSC)\ne 0$