Let $T$ be a bounded linear operator on an infinite dimensional Hilbert space $V$, and let $\sigma(T)$ be the spectrum of $T$. This is the set of complex numbers $\lambda$ such that $T - \lambda 1_V$ does not have a bounded inverse. It is always nonempty and compact.
There are two reasons why a complex number $\lambda$ could be in the spectrum:
$T - \lambda 1_V$ is not injective. In this case, $\lambda$ is called an eigenvalue of $T$.
$T - \lambda 1_V$ is injective, but not surjective.
The case where $T - \lambda 1_V$ is bijective but does not have a continuous inverse never occurs, because of the open mapping theorem.
The set of eigenvalues of $T$ is called the point spectrum. The complement of the point spectrum in $\sigma(T)$ is called the continuous spectrum.
Is there anything that can be said in general about the topology of the point spectrum and continuous spectrum? What are some common conditions on $T$ (like compact, normal etc.) which imply interesting results about these sets?
For example, when $T$ is compact and self adjoint, the continuous spectrum is empty, the spectrum consists of real numbers, and $\sigma(T) - \{0\}$ is discrete.
Suppose $\mathcal{H}$ is a Hilbert space with orthonormal basis $\{ e_{\alpha} \}_{\alpha\in\Lambda}$. You can define a bounded normal operator $L : \mathcal{H}\rightarrow\mathcal{H}$ with $\|L\| \le M$ in such a way that $Le_{\alpha}=\lambda_{\alpha}e_{\alpha}$ for any choice of constants $\lambda_{\alpha}$ for which $|\lambda_{\alpha}|\le M$. The spectrum of $L$ is the closure $\{\lambda_{\alpha}\}^c$ of $\{ \lambda_\alpha \}$. Any point in the set $\{\lambda_\alpha\}$ is an eigenvalue, and any point of the closure $\{\lambda_{\alpha}\}^c$ that is not in this set is in the continuous spectrum. What sets of eigenvalues you can create in this this way depends in part on the cardinality of the index set $\Lambda$. But you can force the spectrum $\{ \lambda_{\alpha}\}^c$ to be any compact subset of $\mathbb{C}$ if $\Lambda$ is countably infinite; the continuous spectrum is $\{\lambda_{\alpha}\}^c\setminus\{\lambda_{\alpha}\}$, and the point spectrum is $\{\lambda_{\alpha} \}$.