Connected and disconnected space [duplicate]

$GL(2 , R) = \{ M | 2 × 2 \& \det M \neq 0\}$ be defined as subset of $\mathbb{R}^4$

For example

Consider the matrix GL(2, R) =

\begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix}

then (1,2,3,4)∈ R4 Accordingly, is the GL(2, R) space a connected space? Explain your answer.

a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets.

If X is a connected space and $f : X \to Y$ is a continuous function, then f(X) is connected.

Edit: My idea was to construct a continuous function from GL(2,R) to R4 and show that f(GL(2,R)) is not connected but I am not reaching a contradiction please help

Solution 1:

Using your result about connectedness if you define $\text{det}:GL(2,\mathbb{R})\to \mathbb{R}$ as the function that sends each matrix to its determinant, then by definition of $GL(2,\mathbb{R})$ you have that $\text{Im}(\text{det})=\mathbb{R}\setminus\lbrace0\rbrace$, which is not connected. Thus, since the determinant is a continuous function (it is equivalent to a polynomial function) we can apply your result and say that $GL(2,\mathbb{R})$ is not connected.