Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X_c := f^{-1} ([0,c])$.

Motivation If $X$ is a closed manifold and $f$ is smooth and $0$ is a regular value, then $X_0$ is a smooth submanifold and $X_c\cong X_0\times [0,1]$ when $c$ is small enough. Consequently, the cohomology of the pair $H^*(X_c, X_0)\simeq (0)$. If $X$ is a closed manifold and $f$ is smooth (without the assumption of $0$ being a regular value)the same still holds even if $X_0$ now can have singularities, indeed we can consider the flow of the normalized gradient of $f$, $\Phi:X\times \mathbb{R}\to X$ which gives a retraction of $X_c$ onto $X_0$.

What can be said in the general case?

Is it still true that there for all $c$ sufficiently small $H^*(X_c, X_0) \simeq (0)$? Does it hold at least when $X$ is a topological manifold?

Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X_c := f^{-1} ([0,c])$. Furthermore, suppose that $X_0 \neq \emptyset$ and $f$ is proper.

Motivating example If $X$ is a closed manifold and $f$ is smooth and $0$ is a regular value, then $X_0$ is a smooth submanifold and $X_c\cong X_0\times [0,1]$ when $c$ is small enough. Consequently, the cohomology of the pair $H^*(X_c, X_0)\simeq (0)$. If $X$ is a closed manifold and $f$ is smooth (without the assumption of $0$ being a regular value)the same still holds even if $X_0$ now can have singularities, indeed we can consider the flow of the normalized gradient of $f$, $\Phi:X\times \mathbb{R}\to X$ which gives a retraction of $X_c$ onto $X_0$.

What can be said in the general case?

Is it still true that there for all $c$ sufficiently small $H^*(X_c, X_0) \simeq (0)$? Does it hold at least when $X$ is a topological manifold?

Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X_c := f^{-1} ([0,c])$.

If $X$ is a closed manifold and $f$ is smooth and $0$ is a regular value, then $X_0$ is a smooth submanifold and $X_c\cong X_0\times [0,1]$ when $c$ is small enough. Consequently, the cohomology of the pair $H^*(X_c, X_0)\simeq (0)$.

If $X$ is a closed manifold and $f$ is smooth (without the assumption of $0$ being a regular value)the same still holds even if $X_0$ now can have singularities, indeed we can consider the flow of the normalized gradient of $f$, $\Phi:X\times \mathbb{R}\to X$ which gives a retraction of $X_c$ onto $X_0$.

What can be said in the general case?

Is it still true that there for all $c$ sufficiently small $H^*(X_c, X_0) \simeq (0)$? Does it hold at least when $X$ is a topological manifold?

Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X_c := f^{-1} ([0,c])$.

Motivation If $X$ is a closed manifold and $f$ is smooth and $0$ is a regular value, then $X_0$ is a smooth submanifold and $X_c\cong X_0\times [0,1]$ when $c$ is small enough. Consequently, the cohomology of the pair $H^*(X_c, X_0)\simeq (0)$. If $X$ is a closed manifold and $f$ is smooth (without the assumption of $0$ being a regular value)the same still holds even if $X_0$ now can have singularities, indeed we can consider the flow of the normalized gradient of $f$, $\Phi:X\times \mathbb{R}\to X$ which gives a retraction of $X_c$ onto $X_0$.

What can be said in the general case?

Is it still true that there for all $c$ sufficiently small $H^*(X_c, X_0) \simeq (0)$? Does it hold at least when $X$ is a topological manifold?